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Modeling Aeolian Dune and Dune Field Evolution
Item Type text; Electronic Dissertation
Authors Diniega, Serina
Publisher The University of Arizona.
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Download date 23/05/2018 01:28:19
Link to Item http://hdl.handle.net/10150/195665
Modeling Aeolian Dune and Dune Field
Evolution
by
Serina Diniega
A Dissertation Submitted to the Faculty of the
GRADUATE INTERDISCIPLINARY PROGRAM IN
APPLIED MATHEMATICS
In Partial Fulfillment of the RequirementsFor the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2 0 1 0
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THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Serina Diniega entitled Modeling Aeolian Dune and Dune Field Evolution and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy ____________________________________________________________Date: 05/12/10 Karl Glasner ____________________________________________________________Date: 05/12/10 Michael Tabor ___________________________________________________________Date: 05/12/10 Kevin Lin ___________________________________________________________Date: 05/12/10 Shane Byrne Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ____________________________________________________________Date: 05/12/10 Dissertation Director: Karl Glasner
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Statement by Author
This dissertation has been submitted in partial fulfillment of requirements for anadvanced degree at The University of Arizona and is deposited in the UniversityLibrary to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,provided that accurate acknowledgment of source is made. Requests for permissionfor extended quotation from or reproduction of this manuscript in whole or in partmay be granted by the head of the major department or the Dean of the GraduateCollege when in his or her judgment the proposed use of the material is in the interestsof scholarship. In all other instances, however, permission must be obtained from theauthor.
Signed: Serina Diniega
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Acknowledgements
First, of course, I thank my advisors. Thank you, Karl, for taking on the dune projectso long ago, for showing me how to be a mathematician within an interdisciplinaryfield, and for teaching me the importance of telling “a good story.” Thank you, Shane,for letting me rope you into being my minor advisor and for always finding the timeand energy to be my sounding board (and emotion-validator) through a wide rangeof discussion topics.
I also thank the rest of my committee: Dr. Tabor, thank you for guiding methroughout my graduate career. Kevin, thank you for being my fourth member!
I heartily thank my friends and family for supporting me through all of the toughtimes. I don’t know that I would have made it to this point without you; I do knowthat I wouldn’t have had this much sanity and self-confidence intact without yourhelp. You all were highly influential as I found my path through the rabbit-hole ofgraduate school and research.
Finally, I thank Abe. Thank you for supplying hugs, food, and footrubs wheneverneeded; for listening to the frustrated rants, excited research plans, and everythingin-between; for encouraging me through the disappointments and for celebrating theaccomplishments. In short: thank you, love, for everything.
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Dedication
I dedicate this dissertation to my family – the most important part of my life and
my primary source of strength and inspiration. In particular, I dedicate this work to
the members of the younger generations – I pray that I will provide assistance and
support to your life’s journey, as was provided to me.
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Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2. Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3. This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. THE PHYSICAL SYSTEM . . . . . . . . . . . . . . . . . . . . . . . 222.1. Dunes and Dune Fields . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1. Dunes forming in unidirectional wind flow . . . . . . 232.1.2. Dune geomorphological terminology . . . . . . . . . 24
2.2. Relevant Processes and Environmental Conditions . . . . . . . . 252.2.1. Sand motion . . . . . . . . . . . . . . . . . . . . . . 262.2.2. Wind velocity profile and shear stress . . . . . . . . 262.2.3. Sand flux . . . . . . . . . . . . . . . . . . . . . . . . 282.2.4. The shadow zone . . . . . . . . . . . . . . . . . . . . 30
3. THE DUNE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1. Model Description . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1. Separation bubble . . . . . . . . . . . . . . . . . . . 333.1.2. Shear stress . . . . . . . . . . . . . . . . . . . . . . . 353.1.3. Saturated sand flux . . . . . . . . . . . . . . . . . . 363.1.4. Sand flux . . . . . . . . . . . . . . . . . . . . . . . . 373.1.5. Mass conservation and avalanching . . . . . . . . . . 383.1.6. Complete system . . . . . . . . . . . . . . . . . . . . 403.1.7. Non-dimensionalization . . . . . . . . . . . . . . . . 403.1.8. Simulation algorithm . . . . . . . . . . . . . . . . . . 42
3.2. Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.1. Linear stability analysis . . . . . . . . . . . . . . . . 433.2.2. Reduced dimensional analysis . . . . . . . . . . . . . 44
4. INFLUENCE OF SAND FLUX . . . . . . . . . . . . . . . . . . . . 494.1. Physical Description . . . . . . . . . . . . . . . . . . . . . . . . 494.2. Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1. The need for a non-zero influx . . . . . . . . . . . . 50
Table of Contents—Continued
7
4.2.2. Sand flux over a dune . . . . . . . . . . . . . . . . . 514.3. Physical Implications . . . . . . . . . . . . . . . . . . . . . . . . 53
5. INFLUENCE OF DUNE INTERACTIONS . . . . . . . . . . . . . 555.1. Physical Description . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1.1. Effects not included in this study . . . . . . . . . . . 565.1.2. First-order effect of topography on shear stress . . . 56
5.2. Study of Dune Collisions . . . . . . . . . . . . . . . . . . . . . . 595.2.1. Defining the interaction function . . . . . . . . . . . 605.2.2. Dependencies of the interaction function . . . . . . . 655.2.3. Implications of the crossover value . . . . . . . . . . 665.2.4. Other studies and their interaction functions . . . . 71
5.3. Field Model Description . . . . . . . . . . . . . . . . . . . . . . 735.3.1. Approximation of single dunes . . . . . . . . . . . . 735.3.2. Interactions and initialization: particle model . . . . 745.3.3. Model assumptions . . . . . . . . . . . . . . . . . . . 755.3.4. Possible field end states . . . . . . . . . . . . . . . . 77
5.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.4.1. Threshold between end states . . . . . . . . . . . . . 795.4.2. Influence of other model parameters . . . . . . . . . 805.4.3. Physical implications . . . . . . . . . . . . . . . . . . 825.4.4. Possible future model improvements . . . . . . . . . 87
6. INFLUENCE OF BEDROCK TOPOGRAPHY . . . . . . . . . . . 906.1. Physical Description . . . . . . . . . . . . . . . . . . . . . . . . 906.2. Influence of Terrain on Isolated Dune Evolution . . . . . . . . . 92
6.2.1. Over a smooth hill . . . . . . . . . . . . . . . . . . . 926.2.2. Over upward and downward steps . . . . . . . . . . 936.2.3. Effect on isolated dune evolution . . . . . . . . . . . 97
6.3. Implications for Field Evolution . . . . . . . . . . . . . . . . . . 1016.3.1. Effect on dune collisions . . . . . . . . . . . . . . . . 1016.3.2. Effect on dune field pattern formation . . . . . . . . 1046.3.3. Inclusion of topography in the dune field model . . . 105
7. AN APPLICATION: MARTIAN POLAR DUNES . . . . . . . . . 1077.1. Physical Description . . . . . . . . . . . . . . . . . . . . . . . . 1087.2. Influence of Reversing Winds . . . . . . . . . . . . . . . . . . . 1107.3. Influence of Diffusive Processes . . . . . . . . . . . . . . . . . . 111
7.3.1. Possible method for estimating the diffusion rate . . 1187.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Table of Contents—Continued
8
7.4.1. Implications for martian dunes . . . . . . . . . . . . 122
8. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238.1. Dune Field Morphology . . . . . . . . . . . . . . . . . . . . . . 1238.2. Dune Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.3. Future Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3.1. Dune nucleation . . . . . . . . . . . . . . . . . . . . 1268.3.2. Evolving intrafield sediment conditions . . . . . . . . 1278.3.3. Process interactions . . . . . . . . . . . . . . . . . . 128
8.4. The Need for Comparative Observations . . . . . . . . . . . . . 130
A. THE NUMERICAL ALGORITHM . . . . . . . . . . . . . . . . . . 132A.1. Avalanching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.2. Separation bubble . . . . . . . . . . . . . . . . . . . . . . . . . 133A.3. Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.3.1. Jackson-Hunt equation . . . . . . . . . . . . . . . . . 135A.4. Sand flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.5. Update the dune profile . . . . . . . . . . . . . . . . . . . . . . 140
B. TWO-DUNE REDUCED DIMENSION MODEL . . . . . . . . . . 141B.1. Two-dune Structure . . . . . . . . . . . . . . . . . . . . . . . . 141B.2. Derivatives of the Two-dune Profile . . . . . . . . . . . . . . . . 143B.3. Isolating Time-derivatives . . . . . . . . . . . . . . . . . . . . . 145
C. SMOLUCHOWSKI DUNE INTERACTION STUDY . . . . . . . 149C.1. Rate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C.2. Output Function . . . . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
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List of Tables
Table 3.1. Terrestrial and martian dune parameters. . . . . . . . . . . . . . 41
Table 5.1. Probability that an average collision will yield similar-sized dunes. 80Table 5.2. Influence of different dune field model components. . . . . . . . . 82
Table 6.1. Control simulation with flat topography. . . . . . . . . . . . . . . 93
Table 7.1. Connections between dune slope and processes. . . . . . . . . . . 120
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List of Figures
Figure 1.1. Interactions between wind, sediment, and bedforms. . . . . . . 15Figure 1.2. Example of patterned martian dune field. . . . . . . . . . . . . 16Figure 1.3. Example of runaway growth within a martian dune field. . . . . 17Figure 1.4. First full dune evolution model results, by Werner (1995). . . . 18Figure 1.5. Dune types in different wind and sand availability regimes. . . . 19
Figure 2.1. Schematic diagram showing dune terminology. . . . . . . . . . . 25Figure 2.2. Sediment transport and dune dynamics. . . . . . . . . . . . . . 27Figure 2.3. Typical grain size distribution within dunes. . . . . . . . . . . . 28Figure 2.4. Shearstress velocity required to move particles. . . . . . . . . . 29Figure 2.5. Lee airflow patterns. . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 3.1. Calculated shear stress over a dune. . . . . . . . . . . . . . . . 36Figure 3.2. Variables computed in continuum dune model. . . . . . . . . . 40Figure 3.3. Steady-state two-dimensional dune profiles. . . . . . . . . . . . 43Figure 3.4. Dune size histograms within an Earth and Mars dune field. . . 45Figure 3.5. Piecewise linear dune profile approximation. . . . . . . . . . . . 46Figure 3.6. Phase diagram showing stoss slope evolution. . . . . . . . . . . 47
Figure 4.1. Different dune dynamics based on sand influx. . . . . . . . . . . 52Figure 4.2. Dune field instability due to sand flux. . . . . . . . . . . . . . . 54
Figure 5.1. Diagram used in estimate of topography-induced shear stress. . 57Figure 5.2. Shear stress perturbation as a function of distance. . . . . . . . 58Figure 5.3. Diagram illustrating a dune collision. . . . . . . . . . . . . . . . 61Figure 5.4. Collision results as a function of dune sizes. . . . . . . . . . . . 62Figure 5.5. Interaction rule, as a function of dune size ratio. . . . . . . . . . 63Figure 5.6. Interaction rules for barchan dune collisions. . . . . . . . . . . . 64Figure 5.7. Interaction functions with varied separation bubble size. . . . . 67Figure 5.8. Example linear interaction functions used in study. . . . . . . . 71Figure 5.9. Limit in dune height found with Pelletier (2009) study. . . . . . 72Figure 5.10. Interaction function found with modified shear stress. . . . . . . 73Figure 5.11. Morphology assumed for point-mass dunes. . . . . . . . . . . . 74Figure 5.12. Model end state as a function of r∗ and σ/M (uniform). . . . . 81Figure 5.13. Model end state as a function of r∗ and σ/M (Gaussian). . . . . 86
Figure 6.1. Bedrock topographies used in study of dune migration. . . . . . 92Figure 6.2. 0.5-high hill-induced changes in dune shape and velocity. . . . . 94Figure 6.3. 1-high hill-induced changes in dune shape and velocity. . . . . . 95Figure 6.4. 5-high hill-induced changes in dune shape and velocity. . . . . . 96Figure 6.5. 0.5-high step-induced changes in dune shape and velocity. . . . 98
List of Figures—Continued
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Figure 6.6. 1-high step-induced changes in dune shape and velocity. . . . . 99Figure 6.7. 2-high step-induced changes in dune shape and velocity. . . . . 100Figure 6.8. Maximal change in dune velocity due to topography. . . . . . . 101Figure 6.9. Distance of dune perturbation due to topography. . . . . . . . . 102Figure 6.10. Mass exchange over flat versus non-flat topography. . . . . . . . 103
Figure 7.1. Example of non-smooth lee slope seen on martian dunes. . . . . 107Figure 7.2. Example of other lee slopes seen on martian dunes. . . . . . . . 108Figure 7.3. Locations of martian dunes with slipface slope break. . . . . . . 109Figure 7.4. Adjustment of dune experiencing wind reversal. . . . . . . . . . 110Figure 7.5. Example of reversing martian dunes. . . . . . . . . . . . . . . . 112Figure 7.6. Dune profiles formed under wind reversals. . . . . . . . . . . . . 113Figure 7.7. Evolution of mean slope in 3:1 wind reversals. . . . . . . . . . . 114Figure 7.8. Mean slope values resulting from wind reversals. . . . . . . . . . 115Figure 7.9. Mean slope values in reversing winds and higher diffusion. . . . 116Figure 7.10. Example diffusive process: CO2 sublimation. . . . . . . . . . . . 117Figure 7.11. Schematic diagram of lee slope measurables. . . . . . . . . . . . 118Figure 7.12. Lee slope measurables on simulated dunes. . . . . . . . . . . . . 119Figure 7.13. Flat-topped Antarctic dune. . . . . . . . . . . . . . . . . . . . . 121Figure 7.14. Flat-topped martian dune. . . . . . . . . . . . . . . . . . . . . . 121
Figure A.1. Slope check calculation in avalanching step. . . . . . . . . . . . 133Figure A.2. Ground check calculation in avalanching step. . . . . . . . . . . 134Figure A.3. Airflow regimes over low hill. . . . . . . . . . . . . . . . . . . . 136Figure A.4. Perturbation velocities over low hill. . . . . . . . . . . . . . . . 139
Figure B.1. Example two-dune structure. . . . . . . . . . . . . . . . . . . . 142Figure B.2. Partial derivatives of the two-dune model. . . . . . . . . . . . . 144Figure B.3. Phase diagram of stoss slope evolution, two-dune model. . . . . 147
Figure C.1. Analytically-derived rate of interaction function. . . . . . . . . 151Figure C.2. Numerically-calculated rate of interaction function. . . . . . . . 152
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Abstract
sand hops and bounces –
see the dunes grow, run, collide –
form the field’s pattern.
Aeolian sand dune morphologies and sizes are strongly connected to the envi-
ronmental context and physical processes active since dune formation. As such, the
patterns and measurable features found within dunes and dune fields can be inter-
preted as records of environmental conditions. Using mathematical models of dune
and dune field evolution, it should be possible to quantitatively predict dune field
dynamics from current conditions or to determine past field conditions based on
present-day observations.
In this dissertation, we focus on the construction and quantitative analysis of a
continuum dune evolution model. We then apply this model towards interpretation
of the formative history of terrestrial and martian dunes and dune fields. Our first
aim is to identify the controls for the characteristic lengthscales seen in patterned
dune fields. Variations in sand flux, binary dune interactions, and topography are
evaluated with respect to evolution of individual dunes. Through the use of both
quantitative and qualitative multiscale models, these results are then extended to
determine the role such processes may play in (de)stabilization of the dune field. We
find that sand flux variations and topography generally destabilize dune fields, while
dune collisions can yield more similarly-sized dunes. We construct and apply a phe-
nomenological macroscale dune evolution model to then quantitatively demonstrate
how dune collisions cause a dune field to evolve into a set of uniformly-sized dunes.
Our second goal is to investigate the influence of reversing winds and polar processes
13
in relation to dune slope and morphology. Using numerical experiments, we investi-
gate possible causes of distinctive morphologies seen in Antarctic and martian polar
dunes. Finally, we discuss possible model extensions and needed observations that
will enable the inclusion of more realistic physical environments in the dune and dune
field evolution models.
By elucidating the qualitative and quantitative connections between environmen-
tal conditions, physical processes, and resultant dune and dune field morphologies,
this research furthers our ability to interpret spacecraft images of dune fields, and
to use present-day observations to improve our understanding of past terrestrial and
martian environments.
14
1. INTRODUCTION
Interactions between wind flow, transported granular sediment, and bedform mor-
phology (Figure 1.1) create aeolian landforms which record contemporaneous envi-
ronmental conditions and processes, over a range of spatial and temporal scales. For
example, the sizes and shapes of terrestrial dunes strongly reflect surface and at-
mospheric conditions throughout their evolution, which can span seasons to decades
and decameters to kilometers (Sauermann et al. 2001, Sauermann et al. 2003, Bourke
et al. 2009). On much larger scales, the spatial distribution of and complexity within
martian and terrestrial dune fields are studied as records of regional conditions over
long timescales (Kocurek & Ewing 2005, Bishop 2007, Hayward et al. 2009).
Within the last few decades, much progress has been made towards quantitatively
connecting environmental conditions with resultant dune morphology and behavior.
This is primarily due to the increased availability of spacecraft images of dune fields
on different planetary surfaces, and the development of a detailed continuum dune
evolution model. The combination of these two advances has greatly aided dune
evolution studies as images of dunes evolving within a range of environmental condi-
tions have allowed for better validation of model assumptions and structures and the
application of these models has improved landform image interpretation.
1.1. Motivation
Dune and dune evolution models have not been able to replicate all observations,
however, which limits their use in connecting dune field morphology with environ-
mental conditions. In particular, an important open question remains regarding the
equilibrium or maximum size of dunes which creates the uniform dune sizes found
15
bedform
evolution
and variation
topography
and slope
effectsenergy
regimeflow
diversion
momentum extraction
bedform
morphology
airflow
(wind)sediment
transport
grain motion
Figure 1.1. Interactions and feedbacks that occur in the evolution of an aeolianbedform system (based on a figure in Walker & Nickling 2002). Aeolian bedformsform over a wide range of scales, from ripples (decimeters to meters wavelength) todunes (decameters to kilometers), to dune fields and mega-dunes.
within many dune fields (such as the crescentic dunes in White Sands National Mon-
ument, New Mexico or the martian dune field shown in Figure 1.2). It is generally
accepted that these dune field patterns are indicative of evolution through a process
of self-organization (Werner 1995, Kocurek & Ewing 2005, Bishop 2007) towards a
steady state.
Not all fields are patterned, however. Some fields, such as the Kelso dune field in
California or the martian dune field shown in Figure 1.3, show signs of a disrupted
initial pattern and scaling, with the majority of dunes being of similar sizes and a
few dunes that are significantly larger. It is an open question if this is due to abrupt
changes in environmental conditions, a slow and natural evolution within the field
conditions or dynamics, or a scale-dependent effect of boundary conditions. Patterns
(or lack thereof) within different fields may also result from localized influences and
from a combination of processes and conditions.
Despite these complications, a quantitative understanding of how dune field pat-
terns can form (or be disrupted) is needed as this directly relates to assumptions
regarding the timescale over which dunes and dune fields record information.
In this research, we evaluate the influence of topography and dune collisions on
16
���
Figure 1.2. This image (HiRISE PSP 008628 2515) shows a martian dune fieldwithin a crater with two dominant dune sizes (200m and 600m), except for near theboundaries of the field (where topography is probably influencing dune evolution).Such pattern superposition can occur through multiple generations of dune construc-tion and evolution, where in each generation the dune field tends towards a simplepattern through dune interaction (Kocurek & Ewing 2005).
dune and dune field evolution. We focus on the identification of characteristic length-
scales, and related environmental or dynamic controls, that may yield a maximum
dune size. The conditions under which a dune field will not form similarly-sized dunes,
but instead will contain a few that are significantly larger, are also explored.
1.2. Previous Studies
The first rigorous studies of aeolian bedforms were completed by Bagnold (1941) who
showed that aeolian sand ripples evolve into bedforms of characteristic spacing and
heights, and that these lengthscales are controlled in a consistent manner by grain
size and wind conditions. As dunes exhibit similar morphologies and behaviors as
ripples, and are formed by the same physical processes, it is assumed that they can
be modeled with a similar method. The first full formation and evolution models
of dunes utilized a simple cellular automata model of wind-transported sand slabs
where each transported slab would move a discrete distance during a time step; this
17
Figure 1.3. This image (HiRISE PSP 007172 2570) shows a martian dune fieldwhere runaway growth appears to be occuring. Many small, uniform barchandunes (200m wide) populate this area, interspersed with a small number of largemegabarchans (large, central one is ∼1040m wide). All dunes have left-facing slip-faces, suggesting that they formed under the same environmental conditions and were(are?) migrating towards the left. The megabarchans have sinuous crests and mul-tiple slipfaces on the windward slope, suggesting that these dunes formed throughcollision of the barchans (a hypothesis also put forward by Bourke & Balme (2008)).The areas downwind of the two megabarchans shown are mostly devoid of other dunestructures, implying that the collisions are predominantly resulting in coalescence.There are no apparent topographical or environmental reasons for the size disparitybetween the megabarchans and the barchan dunes.
saltation-type motion can be considered the large-scale and average motion of an
accumulation of sand (Werner 1995, Momiji et al. 2000, Bishop et al. 2002). These
minimal phenomenological models, surprisingly, yielded bedforms with dune shapes
that varied depending on sediment supply and wind variability (Figure 1.4), as ob-
served in nature (Figure 1.5).
Numerical experiments with these discrete models aimed to identify the influence
of non-linear effects, such as wind speedup and associated sand flux variations on the
upwind slopes. Additional studies attempted to identify characteristic lengthscales
within dune fields. Recent work by Pelletier (2009) argued that dunes saturate in size
due to feedback between landform-induced surface roughness and saltation flux. Due
to the discrete nature of these models, however, it is difficult to quantitatively connect
the influence of specific environmental conditions on details of dune morphology, size,
and behavior.
18
A B C
Figure 1.4. Dune fields created with a cellular automata model, under differingsequences of transport direction (indicated with arrows). Initial topography in allsimulations was random in morphology. A: Simulated barchan dunes form underunidirectional wind. B: Simulated linear dunes form under bi-directional winds. C:Simulated star dunes form under four wind directions. Images taken from Werner(1995).
Continuum models of dune evolution were also developed which considered sand
flux over topography and then related that flux to changes in topography (Howard
et al. 1978, Stam 1996, Sauermann et al. 2001). The derivation and analysis of model
equations allowed model parameters (and environmental conditions) to be quantita-
tively connected with characteristic landform scales and morphologies, yielding infor-
mation about the minimum size of dunes (Kroy et al. 2002) and the initial growth
of dunes (Andreotti et al. 2002b). However, these continuum models did not yield
information about the maximum dune size. In fact, numerical simulations showed
that dunes will grow without bound unless the sediment supply is limited. Based on
these dynamics, Hersen et al. (2004) showed that if dunes interact with each other
only through sediment flux, then dunes within that field will have no characteris-
tic maximum lengthscale. Instead, dunes will coalesce, perpetually yielding smaller
numbers of larger dunes.
It appears that the largest dunes may be limited in size due to resonances with the
atmospheric boundary layer (Andreotti et al. 2009). However, this does not account
19
Figure 1.5. Diagram showing the dune types/shapes that form in different windand sand availability regimes. Image is from Bishop et al. (2002).
for the seemingly limited sizes and spacings of smaller dunes. One possibility is that
stochastic processes, such as variations in wind strength and direction, may play a role
in the stabilization of dune fields by preferentially destabilizing large dunes (Hersen
et al. 2004, Elbelrhiti et al. 2005, Elbelrhiti et al. 2008). However, such processes,
being stochastic, are difficult to quantify and to incorporate into generic analysis
or simulation of dune fields. A study by Werner (1999) attempted to do this, by
estimating rates of bedform defect merging and relating this to average dune sizes.
Another mechanism that has been proposed to regulate dune size and spacing, and
that depends on sand and dune dynamics broadly common to all dune fields, is dune
collision. When dunes collide, sand is redistributed between the dunes. As long as the
net sand exchange is from the large to the small dune, it is possible for a dune field
to evolve into a patterned structure after many collisions (Hersen & Douady 2005).
Some studies have considered the evolution of dune fields through collisions between
dunes (Lima et al. 2002, Parteli & Herrmann 2003, Lee et al. 2005) and have created
fields containing similarly-sized dunes. However, these models generally rely on ill-
20
defined assumptions and model-specific parameters, making it difficult to generalize
their results.
1.3. This Work
In this dissertation, we aim to explore the influence of specific environmental condi-
tions (i.e., sand flux, topography, and reversing winds) and dynamical processes (i.e.,
binary interactions, aeolian/cold climate process interactions), and to quantitatively
connect these influences to dune and dune field morphology. These studies have relied
on an iterative combination of model development, analysis, and numerical simula-
tion. Generally, models have been run in non-dimensional space to view the dynamics
and morphological trends without bias from environment-specific parameters. The
scales are then reintroduced for comparison with observed terrestrial and martian
dunes and dune fields.
In Chapter 2, the physical processes and environmental conditions that create
dunes and dune fields are reviewed. These processes and environmental conditions
are discussed again in Chapter 3 in the form of equations and parameters within the
dune evolution model. This section also includes analysis and non-dimensionalization
of the dune evolution model equations.
In Chapters 4 through 6, we investigate the influence of processes and environ-
mental conditions on dune size regulation. In Chapter 4, we consider the influence
of sand flux on and between dunes. In Chapter 5, we consider the influence of in-
teractions between dunes. In particular, we couple the dune model with a simple,
phenomenological dune field model to evaluate the effect dune collisions will have
on dune size evolution within a field; most of this content is published in Diniega
et al. (2010). Finally, in Chapter 6 we consider the influence of topography on dune
migration, and shape and size stability. We do this by evaluating the effect a small
hill or step will have on isolated dune evolution and a binary dune collision, and then
21
extending those results to dune field evolution.
In Chapter 7, we consider an application of the dune evolution model, which is
extended to include reversing winds and cold climate processes. We present prelimi-
nary investigations of the influence these additional processes will have on dune slope
evolution. In particular, we aim to understand the formation of distinctive lee slope
morphology observed within Antarctic and martian polar dunes.
22
2. THE PHYSICAL SYSTEM
Aeolian dunes develop whenever there is a source of granular material, wind of suf-
ficient strength to move this material, and an “obstacle” to induce initial accumula-
tion. These landforms have been found on Earth, Mars (e.g., Greeley et al. 2003),
Venus (e.g., Greeley et al. 1992), and even the Saturnian satellite Titan (e.g., Lorenz
et al. 2006). Dunes can also form underwater and are similar in morphology and be-
havior to aeolian dunes (Endo, Kobu & Sunamura 2004) after appropriate rescaling
(Claudin & Andreotti 2006). In this study, however, we focus exclusively on aeolian
dunes.
In this chapter, we briefly explain how dunes form and identify important physical
processes and environmental conditions. This will help motivate the model we use to
explore dune and dune field dynamics, which is discussed in Chapter 3.
2.1. Dunes and Dune Fields
Dunes are classified based on their large-scale morphology, which has been observed
to be strongly correlated with two environmental conditions: the amount of sand
available and the variation in wind direction (Figure 1.5). This correspondence indi-
cates that sand flux dynamics and localized sediment and wind conditions are highly
influential in determining the shape of dunes, and that these specific dune shapes
are “attractor” or equilibrium states (Werner 1995, Hesp & Hastings 1998, Andreotti
et al. 2002a). This allows dune morphology to be used to infer the environmental con-
ditions during dune formation. Conversely, a lack of correlation between the present
environment and a dune shape is indicative of changes in the wind and/or sediment
parameters.
However, many dunes have complex shapes and dune fields may consist of super-
23
imposed dune forms. To reverse-engineer the environmental conditions throughout
the evolutionary age of the dune field from complicated dune and dune field morphol-
ogy, quantitative models are needed to determine exactly how a dune’s size and shape
will evolve. Such models are also essential in determining how a dune will respond
to changes in the environment and to estimate the temporal and spatial scales of
dune adjustment. Finally, models are needed to decouple the effects of changes in
localized environmental conditions (e.g., due to natural evolution of sediment supply
or seasonal wind variation) versus general dynamic processes (e.g., due to variations
in sand flux between dunes as dunes grow and/or collide).
2.1.1. Dunes forming in unidirectional wind flow
The dune and dune field models considered in this study are two-dimensional. This
restricts their application to dunes which form in unidirectional wind flow, and ne-
glects processes and forces that occur transverse to the wind direction. In natural
environments, even strongly unidirectional wind flow will have some variation, and
this may need to be considered when examining detailed dune geomorphology. This
study focuses only on general trends and scalings, however, which are assumed to be
relatively invariant with respect to slight wind variation and sand flux in the third
dimension (when averaged over sufficient time). Three-dimensional models have used
to examine more complicated dune forms (Parteli, Schatz & Herrmann 2005, Parteli
& Herrmann 2007, Reffet et al. 2010), but these models are usually based on sim-
ple extensions of the two-dimensional model (such as two-dimensional slices, coupled
through transverse diffusion); it remains to be shown that these simple models ade-
quately reflect the physical three-dimensional effects and processes.
Two dune types form in unidirectional winds: transverse dunes and barchan dunes
(Figure 1.5). Transverse dunes are extensive, linear dunes with crest lines perpen-
dicular to the wind direction; the dominant wind direction can be determined by
24
locating the steeper slope (which faces downwind). These dunes form in regions of
high sand supply, with small or non-existent sand-free interdunal areas (sand cov-
erage of ≤ 10%; Lancaster 1995). Transverse dunes are found in 40% of terrestrial
sand seas and over a majority of martian sand seas (Wiggs 2001, Schwammle &
Herrmann 2004). A reversing dune is a transverse dune that evolves under winds
that periodically change direction 180o. Three-dimensional simulations of transverse
dunes produce translationally invariant dunes (Schwammle & Herrmann 2004), so
two-dimensional simulations are generally assumed to be sufficient (e.g., unless de-
fects occurring along a dune crestline are of interest; Werner 1999).
Barchan dunes, also known as crescent dunes, are often the focus of dune studies
as they are one of the smallest dune forms, are often easily accessible by car, and form
under “simple” conditions of low sediment supply and unidirectional wind (Andreotti
et al. 2002a). These dunes are isolated bedforms, with large sand-free interdune
areas (Lancaster 1995). Dominant wind direction can be determined easily from their
shape, as their horns and their steepest slope point downwind. These dunes have an
intrinsically three-dimensional shape (Schwammle & Herrmann 2005), which causes
important differences in sandflux (Hersen 2004) and dune collisions (Gay 1999, Endo,
Taniguchi & Katsuki 2004), so care must be taken in applying two-dimensional model
results towards the interpretation of these three-dimensional landforms.
2.1.2. Dune geomorphological terminology
The dune’s stoss slope is the upwind (or windward) slope, and the lee slope faces
downwind. The dune’s brink is the location on the lee slope where flow separation
occurs and gravity-driven avalanching becomes the dominant process, causing the
slope to abruptly steepen to the angle of repose (∼ 34o). The dune surface that is at
the angle of repose is called a slipface; this surface is always oriented downwind and
rapidly adjusts to environmental changes, so is the best indicator of the recent wind
25
direction (Bagnold 1941). The dune crest is the highest point on the dune, which
does not always coincide with the brink (Schwammle & Herrmann 2005, Parteli,
Schwammle, Herrmann, Monteiro & Maia 2005, Schatz & Herrmann 2006). These
terms are illustrated in Figure 2.1.
���������������
��� ���
����������
����������
�������������
Figure 2.1. Schematic diagram illustrating some dune geomorphology terminology.
2.2. Relevant Processes and Environmental Conditions
When out in the field, especially under strong wind conditions, it is apparent that the
sand transport is complicated and chaotic. It involves interaction between the (gusty)
wind and surficial grains (through wind-exerted shear stress), and between moving
grains and the stationary grains/ground (through granular impacts). Since Bagnold
(1941), much work has gone into simplifying and averaging these small-scale and fast
interactions into quantitative and simple relations between average wind strength and
sand flux (e.g., Fryberger 1979, Iversen & Rasmussen 1999, Sorensen 2004). Other
studies have then aimed at understanding how sand flux relates to dune evolution
(e.g., Howard et al. 1978, van Dijk et al. 1999, Sauermann et al. 2001). Here we
briefly outline dominant processes and conditions; we will consider these again in the
next chapter as we construct the dune model.
26
2.2.1. Sand motion
Sand is transported primarily through interactions between the wind and the ground
surface. As the wind blows over the sand surface, different sized particles are trans-
ported different distances via several processes (Figure 2.2), resulting in a natural
size-sorting within dunes.
Dunes are composed of sand particles (70− 500μm; Figure 2.3); in this context,
sand defines a grain of a specific size (versus a particular composition, etc.) which has
the lowest wind-exerted shear stress threshold for motion (Figure 2.4). This natural
size-sorting results from the difficulty in moving grains of other sizes: larger particles
are more difficult to move due to their higher mass, and smaller particles are more
difficult to move due to higher intergrain cohesion (Bagnold 1941). Sand particles are
lifted by the wind, but are too large to be kept aloft so are moved downwind via short
jumps, in a motion called saltation. Impacting saltating grains will impart energy
to the ground surface, resulting in additional particles moving in small, randomly-
oriented ballistic jumps, in a motion called reptation (Lancaster 1995).
Other types of motion (suspension of smaller particles and creep of larger par-
ticles) are neglected in this dune evolution model, as non-sand-sized particles are
more difficult to move. Additionally, once small grains (dust or clay) are suspended,
they can be transported distances significantly larger than dunes or dune fields (Uno
et al. 2009), and thus are effectively removed from the system.
2.2.2. Wind velocity profile and shear stress
The wind’s shear stress velocity (u∗) quantifies the “strength” of the wind: the shear
stress that it exerts upon the ground (u∗ =√
τ/ρair, where τ is a scalar – the term
within the shear stress tensor that refers to the vertical flux of momentum caused
by the horizontal component of stress (σxz)). Over a flat, non-vegetated plane and
in conditions of neutral atmospheric stability, u∗ is independent of height. It can
27
suspension
saltationreptation
creeptransport of
grains up stoss slopefree-fallon lee slope
wind direction
Figure 2.2. On the left are schematic illustrations of sediment transport processes.Dunes are primarily made up of sand-sized particles (diameter ∼ 100μm), which aretransported via saltation and reptation. On a dune (decameters to a kilometer inlength), grains are transported over the brink of the dune by the wind. If the duneis sufficiently large to shelter the lee region and cause airflow separation (the shadowzone), then these grains free-fall onto the upper surface of the lee slope. As this regionsteepens beyond the angle of repose, small avalanches transport material down-slope– thus, creating a slipface at the angle of repose. These processes give the dune itsdistinctive, asymmetric shape and move the dune forward.
be related to the wind velocity at a height z, however, via the Prandtl-von Karman
model:
u(z) =u∗κ
lnz
z0
, (2.1)
where κ is von Karman’s constant (∼ 0.4), and z0 is the aerodynamic roughness
length. It is important to note that in natural conditions, such as unsteady winds or
sediment transport, the measured wind velocity profiles do not correspond with the
log-linear model close to the surface. However, this model is considered sufficiently
accurate over long temporal and spatial averaging (Walker & Nickling 2002).
The log-linear model, where u∗ and τ are constants in space and time, is also
invalid within an internal boundary layer over large topography, as compression and
acceleration of airflow will alter the shear stress exerted (Frank & Kocurek 1996a):
decreasing slightly at the base of a hill due to flow stagnation, and then increasing
(up to doubling in magnitude) as it moves upslope towards the crest (Walker &
28
Figure 2.3. Diagram illustrating the sharply-peaked and narrowly-supported grainsize distribution found within terrestrial dunes (image taken from Besler 2008) thatforms due to natural process-driven grain size-sorting. As stated by Besler (2008),“the granulometeric distribution in an aeolian deposit is not a random or an ephemeralproperty, but represents a proper characteristic of the adjustment to the depositionalenvironment.”
Nickling 2002, Walker & Nickling 2003). This can be corrected by using a modified
shear stress (τ = τ0(1 + τ )) where the correction factor (τ ) is location- and slope-
dependent (discussed more in subsection 3.1.2).
2.2.3. Sand flux
As the wind blows over a sandy surface, it exerts streamwise shear stress (Walker &
Nickling 2002) and imparts momentum to grains at the surface, initiating a sand flux
(transported via saltation and reptation; Figure 2.2). This sand flux will increase until
the wind-borne momentum is just enough to maintain the sand flux (Owen 1964).
Since Bagnold (1941), much work has been done to quantify this saturated sand
flux (also called the sand drift potential). Bagnold derived and empirically verified
29
Figure 2.4. Diagram showing the shearstress velocity (u∗) needed to move particlesof different diameters due to wind-exerted shear stress. This curve has been deter-mined theoretically and empirically on the Earth, and it is assumed that a similarcurve will be found on other planets. Note that the shown relation (based on a studyby Iversen & White 1982) predicts that martian dune sand should be coarser thanterrestrial dune sand, which is consistent with estimates of average grain size fromthermal measurements (Edgett & Christensen 1991).
that sand flux is proportional to u3∗. Other studies have refined these equations
through laboratory and theoretical studies (e.g., Lancaster 1995, Bullard 1997, Ni
et al. 2004, Sorensen 2004, Duran & Herrmann 2006), but a roughly cubic relation
with the shear stress velocity is consistently found and will be used in the remainder
of this study.
It is important to note, however, that the relationship between wind speed and
sand flux is discontinuous (Lettau & Lettau 1978, Fryberger 1979) as grains are
not moved unless the shear stress exerted on the grains by the wind is above some
threshhold (τt, the fluid threshold in Bagnold 1941). For example, a system of sand
grains with a diameter of 250μm requires a minimum friction velocity of ∼ 0.28m/s
(Sauermann et al. 2001) to move the grains (on Earth). Generally, terrestrial modeling
studies assume that the wind speed is sufficiently above the threshold amount that
30
this is not a concern. When modeling martian dune evolution, this restriction may
become more important, as discussed in Parteli & Herrmann (2007) and this study
(Chapter 7).
An additional complication is that the windspeed threshold for sustaining grain
motion (the impact threshold in Bagnold 1941) is less than τt. This suggests that
although the initial ability of the wind to transport sand may depend primarily on
higher gust velocities, sustained motion may depend on a (much) lower average ve-
locity. This difference is especially important on Mars, where the ratio between the
two thresholds is ∼ 0.1 (Kok 2010). It has been proposed that this hysteresis effect
has important implications for Mars (Almeida et al. 2008); measurements of wind
speed imply that the wind speed may rarely be above the fluid threshold (Greeley
et al. 1980, Parteli & Herrmann 2007) which is inconsistent with observations of salta-
tion (Sullivan et al. 2008). On the Earth, the two thresholds are much closer together
(ratio ∼ 0.8) so this effect can be neglected in modeling studies (Kok 2010).
2.2.4. The shadow zone
As the wind passes over a dune brink, the (previously compressed and accelerated)
airflow expands and decelerates, generating a drop in surface shear stress. This de-
crease in shear stress can be extreme if the change is slope is sufficiently abrupt; as
the airflow passes the dune’s brink, it can separate from the topography and create
a region of low pressure, generally containing a weak recirculation flow (Figure 2.5;
Frank & Kocurek (1996b), Walker & Nickling (2002)). This recirculation flow is gen-
erally too weak to move much sand, so we approximate it as a zone of stagnant air
(Figure 2.2). Grains transported over the brink into this shadow zone simply fall
directly down onto the top few meters of the slipface of the dune, piling up until the
angle of repose is exceeded and avalanching occurs (Lancaster 1995).
It was originally assumed that the extent of the shadow zone would correspond to
31
the distance necessary for flow to return to far-field velocity values. However, wind
tunnel (Walker & Nickling 2002) and numerical modeling (Parsons et al. 2004) ex-
periments indicate that a horizontal distance of tens of the upwind dune’s height is
needed for full boundary layer recovery, which is much larger than the observed spac-
ing between dunes (4-10 times the upwind dune’s height (Lancaster 1995)). Instead,
this distance correlates roughly to the flow reattachment distance (Frank & Kocurek
1996b, Walker & Nickling 2002, Parsons et al. 2004, Schatz & Herrmann 2006) and
it is now thought that full flow recovery may never occur over closely-spaced dunes
(Walker & Nickling 2003). Implications of flow differences over closely-spaced dunes
versus an isolated dune will be examined in more detail in subsection 5.1.2.
Figure 2.5. Schematic diagram showing flow separation as the wind flows over thedune brink, forming a recirculation zone, followed by the reattached flow. Image isfrom Walker & Nickling (2002).
Beyond the shadow zone, observations indicate that rapid erosion will occur as the
wind is undersaturated and accelerating as it recovers (Fryberger et al. 1984, Baddock
et al. 2007).
32
3. THE DUNE MODEL
The seminal work of Sauermann et al. (2001) is the basis for most current continuum
models of the formation and evolution of sand dunes. This study constructed a two-
dimensional model which considered the evolution of an wind-driven sand flux layer
over a dune, and mass exchanges between this layer and the dune itself. Its main
improvement over previous models (e.g., Stam 1996) was the inclusion of a spatial
delay (the saturation length) over which the sand flux evolved toward the carrying
capacity of the wind. This spatial delay had first been observed and measured by
Bagnold (1941) and had been discussed in various modeling studies (e.g., Howard
et al. 1978), but was not incorporated into previous models. Simulation and analysis
have proven the value of including this spatial delay in the continuum dune evolution
model, as it fixes the unphysical anchoring/lengthening of the foot of the dune seen
in previous models and provides a characteristic length scale related to minimal dune
size (Andreotti et al. 2002b, Claudin & Andreotti 2006). Later refinements considered
linearized versions of the equations (Kroy et al. 2002, Andreotti et al. 2002b) as well
as extension to 3-dimensions (Hersen 2004).
Such dune evolution models have been used to study the influences of wind and
sand flux on details of dune morphology (Herrmann et al. 2005, Parteli et al. 2006) and
the influence of vegetation and induration in stabilizing dunes (Herrmann et al. 2008).
Additionally, many studies have examined the characteristic scaling of dune formation
and evolution in different environments, such as underwater or on Mars (e.g. Claudin
& Andreotti 2006, Parteli & Herrmann 2007).
In this chapter, we outline the continuum dune evolution model used in our studies.
We also analyze model equations to determine appropriate model parameters and
characteristic dune size scales.
33
3.1. Model Description
One drawback in the use of a continuum model is the large number of constitutive
equations and environmental parameters that must be specified. In developing our
own version of the two-dimensional continuum dune model, we have endeavored to
identify the ingredients which have a significant quantitative effect on results. With
respect to these ingredients (which are outlined below and illustrated in Figure 3.2),
our model deviates little from assumptions used in other continuum dune models. As
in those models, we evolve two layers of sand: the moving sand volume flux q(x, t)
and dune height h(x, t).
3.1.1. Separation bubble
The profile of the near-surface airflow is a function of topography, and takes into
account airflow separation in regions downwind of sharp changes in slope (the shadow
zone described in subsection 2.2.4). Only weak recirculation occurs in these regions
(Frank & Kocurek 1996b, Walker & Nickling 2002), so sand free-falls onto the lee
slope, and then avalanches down, forming a slip face (Lancaster 1995). The upper-
boundary of the shadow zone is the separation bubble, which extends downwind from
the brink until the flow reattaches to the topography and aeolian sand transport
resumes (Sauermann et al. 2001, Walker & Nickling 2003).
The precise details of flow separation can only be found through fluid dynamic
calculations over the detailed dune topography, which is not well described and which
evolves in time. Fortunately, it appears sufficient to phenomenologically approximate
the separation bubble in calculations of wind-exerted shear stress upon the dune’s
stoss slope (Sauermann et al. 2003). We take this route in our continuum model
and base our separation bubble shape on studies of airflow over simple dune-like
geometries (Schatz & Herrmann 2006). We require only that:
• the separation bubble must match at the dune crest with the value and slope
34
of the topography (C1), and
• the separation bubble is scale-invariant, so will have fixed aspect ratio.
Past studies (e.g. Kroy et al. 2002) use a cubic polynomial as that is a simple
functional that fits the requirements. In this model, we calculate the separation
bubble (si describes the profile of the separation bubble extending downwind from
the ith dune) as the upper-right portion of an ellipse with an semiminor to semimajor
axis ratio of β = 6.5. This is based on the results of Schatz & Herrmann (2006), where
CFD models were used to calculate actual streamlines over dune-type obstacles, and
the streamline coming off the dune brink was fit to an equation.
We also conducted experiments with different types of functions for the separation
bubble (parabolic, cubic, circular, elliptical functions). We found that the exact form
of the function did not greatly change the calculated shear stress on the dune’s stoss-
slope as long it was sufficiently smooth (C1) when paired with the original dune
topography at the dune brink. One way in which the exact functional form of the
separation bubble does matter is that it provides a “separation constraint” on the
topography, as we define airflow separation as occuring when hi < si (equivalently,
when the dune slope hx extending downwind from the brink is steeper than the slope
of si).
The overall separation bubble function s is defined then as a compilation of the
separation bubble found over each individual dune:
s = max(h, si).
Note that as si is only defined downwind of the ith dune, the dune topography and the
streamline always coincide on the upwind slope of the dune (si = hi, excluding any
area sheltered by an upwind dune’s separation bubble); to the lee of a dune, si ≥ hi.
35
3.1.2. Shear stress
As described in subsection 2.2.2, the wind-exerted shear stress depends on the far-
field strength of the wind (τ0) and the topography-induced airflow compression (τ),
such that
τ = τ0(1 + τ ). (3.1)
Over a dune, τ is generally calculated through the use of boundary layer equations
derived for airflow over a gently sloping, symmetrical hill (Jackson & Hunt 1975, Weng
et al. 1991) involving a weighted-integral of the height gradient over the region and
the local gradient itself:
τ = A
∫∞
−∞
sx(x− ξ)
πξdξ + Bsx(x). (3.2)
Note the smooth airflow profile (s) is used to account for airflow separation; this
has the benefit of avoiding numerical difficulties as the dune topography (h) is not
differentiable at the brink.
The parameters A and B are generally treated as phenomenological parameters;
physically, they relate to the logarithmic ratio between the dune and its surface
roughness (discussed in Appendix A). Most studies assume that surface roughness
depends primarily on sand grain diameter; in this case, A and B can be treated as
constants, as the dunes change in size by a factor of 102 maximally, and the sand grains
are≥ 107 times smaller than the smallest dune. A study by Pelletier (2009) considered
that the effective surface roughness will depend on the dominant landform wavelength
within a landscape: as ripples mature, they will dictate the surface roughness (not
the grains), and so on. We have experimented with this approach, but did not find a
large change in dune and dune field evolution.
Generally, we use A = 4, B = 1 in this study (Kroy et al. 2002). These expressions
produce an acceptable approximation of the shear stress exerted along the stoss slope
of an isolated dune (Figure 3.1; Sauermann et al. 2003). In the shadow zone to the
36
lee of the dune, the near-surface wind is negligible so the shear stress is generally
assumed to be zero and the sand flux simply decreases exponentially.
Figure 3.1. Circles show the measured shear velocity (u∗) over a dune, normalizedby the far-field shear stress (u∗,0); these measurements show good agreement withthe shear stress predicted by the Jackson-Hunt equation (solid line) over an isolateddune. To provide spatial context, the dune’s profile is also shown (crosses, with heightnormalized by crest height: h/H). Image taken from Sauermann et al. (2003).
3.1.3. Saturated sand flux
The wind’s shear stress is used to calculate the saturated sand volume flux, which
is the equilibrium volume of sand the wind could move if the wind velocity and
topography were homogenous. Equivalently, this is the sand flux that can be just
maintained by the wind-borne momentum (Owen 1964).
To calculate the saturated sand flux (qs), we use Bagnold’s relation (qs ∼ τ 3/2;
Bagnold 1941), which is a valid approximation as long as τ is sufficiently higher than
the threshhold amount needed to initiate sand movement. We consider the simple
37
linearized relationship:
qs = (τ0(1 + τ ))3/2 (3.3)
∼ τ3/20 (1 +
3
2τ) (3.4)
where the shear stress has been divided into a constant far-field parameter (τ0, the
shear stress exerted by a constant-velocity wind over a flat plane), and a topography-
dependent perturbation term (τ (x, t), defined in subsection 3.1.2).
3.1.4. Sand flux
Due to the discrete nature of sand transport and the complex interactions of saltating
and reptating grains, the actual sand flux evolves after a downwind spatial delay
(called the saturation length) in comparison to the saturated flux:
qx =
⎧⎪⎨⎪⎩qs−q
lsh > ε and h = s
0 h < ε and h = s−qls
h < s.
(3.5)
In these equations, ls is the length over which the sand flux catches up to the saturated
sand flux. Physically, it depends on the distance saltating/reptating sands bounce and
on the ratio of saltating to reptating grains (Sauermann et al. 2001). The constraints
on the different RHS cases correspond to several physical constraints:
• When airflow separation occurs (h < s; only on the leeside of the dune), then
at the surface of the dune there is no effective wind flow and sand free-falls onto
the lee slope.
• The limit ε is the height of sand above some “bedrock” layer (h = 0) below which
we assume h ∼ 0 and no erosion can occur. Thus, when this layer is exposed
(h < ε), qx is constrained to be non-positive. Additionally, if the wind is exerting
shear stress upon the surface (h = s), then it is assumed that moving grains
38
will experience elastic collisions with the hard surface and continue moving (so
qx = 0).
It is important to note that the sandflux depends only on the dune and airflow
profile (through the shear stress calculations), not on sandflux values found in previous
timesteps. This assumption of “instantaneous” sandflux adjustment is based on the
large difference in timescales over which the sand flow and the dune evolve; individual
sand grains migrate tens of centimeters per second and dunes migrate tens of meters
per year (a 106 difference).
3.1.5. Mass conservation and avalanching
Finally, we present the equation that couples our two layers of sand by relating the
sand volume flux (q) to changes in dune height (h). This is a simple mass balance
equation (also known as the Exner equation):
ht = -qx. (3.6)
In our simulations, we use the modified equation:
ht = -qx + (D(hx)hx)x, (3.7)
where the added diffusion term accounts for avalanches. In most other studies
(e.g. Sauermann et al. 2001), avalanching (within each time step) was accounted for
through a separate BCRE-type (Bouchaud et al. 1994) diffusion model. In this work,
the diffusion has been included directly in the model equation to simplify analysis
and numerical simulation.
Avalanches occur only when the dune slope (hx) exceeds the angle of repose (a
maximum slope characteristic to piled cohesionless granular material, ∼ 34o). This
is accounted for by letting the diffusion coeficient D vary as a function of hx. When
the angle of repose is reached or exceeded, D is chosen sufficiently large that the
39
timescale of avalanching dominates all other timescales. When hx is smaller than the
angle of repose, D is chosen small enough so as to operate over a very long timescale,
reflecting small-scale smoothing processes such as turbulent air flow and the random
orientations of reptating grains.
Both analysis and simulation testing were employed to select the values of D(hx).
The Peclet number of the system reflects the relative timescales of diffusion versus
advection, and can be computed as:
Pe = LV/D.
We are interested in dune lee slope dynamics, so L is the length of the lee slope (∼ H ,
where H is the dune crest height) and V is the migration velocity of the dune. A
simple geometric argument shows that V ∼ Q/H, where Q is the sandflux over the
dune crest (Bagnold 1941). Thus:
Pe ∼ Q/D. (3.8)
Diffusion (avalanching) should dominate when the dune slope exceeds the angle of
repose, so we choose D ∼ Q (causing Pe to be small). D is chosen to be two orders
of magnitude smaller than Q otherwise.
In Section 7.3, we will explore the effect and physical significance of this approxi-
mation in more depth.
40
3.1.6. Complete system
The complete and closed system of equations is as follows:
ht = −qx + (D(hx)hx)x (3.9)
qx =
⎧⎪⎨⎪⎩qs−q
lsh > ε and h = s
0 h < ε and h = s−qls
h < s.
(3.10)
qs = Cqτ3/20 (1 +
3
2τ ) (3.11)
τ = A
∫∞
−∞
sx(x− ξ)
πξdξ + Bsx (3.12)
where s is as defined in subsection 3.1.1. Typical parameters used in terrestrial and
martian dune simulations are given in Table 3.1.
dune
actual sand flux
saturatedsand flux
separationbubble
shadowzone
wind
Figure 3.2. Schematic diagram showing the variables that are computed in a contin-uum dune model: dune topography → separation bubble/shadow zone → saturatedsand flux (via shear stress calculation) → actual sand flux → dune topography. Thisdiagram is the result of a simulation run, with the vertical scale of each parameterexaggerated for clearer superposition.
3.1.7. Non-dimensionalization
There are two natural length scales in these equations: the minimum sand depth
at which saltating grains will not be influenced by the underlying bedrock (ε) and
41
Earth Marssaturation length (ls) [m] 1 13wind velocity threshold for saltation (u∗,th) 0.28 1.5
far-field saturation sand flux (qs,∞ = Cqτ3/20 ) at 1.5u∗,th [m2/s] 2e-3 4e-5
diffusion coefficient when avalanches occur (D(hx ≥ 34o)) [m2/s] 1e-2 1e-5diffusion coefficient otherwise (D(hx < 34o)) [m2/s] 1e-4 1e-7lower limit on mobile sand (ε) [m] 1e-3 1e-3
Table 3.1. Typical parameters used in terrestrial and martian dune simulations,based on values found in Sauermann et al. (2001), Parteli & Herrmann (2007). Thevalue for ε was approximated as 10 times the typical dune sand diameter, whichappears comparable between Earth and Mars (Claudin & Andreotti 2006, Parteli &Herrmann 2007).
the saturation length (ls). As ε ∼ d � ls (where d is a typical grain diameter),
the minimal sand depth should be a small value even in the nondimensional model.
Thus, ls is taken as the characteristic lengthscale. The natural scale for the flux is
the unperturbed/flat-plane saturation flux = Cqτ3/20 .
This allows us to define a characteristic time of l2s/(Cqτ3/20 ), leaving just a few
nondimensional constants:
• the parameters in our stress perturbation equation (A and B), which are un-
changed.
• a new lower limit on h (ε/ls → ε),
• and a non-dimensional diffusion coefficient (D(hx)
τ3/2
0
→ D(hx)) which is the inverse
Peclet number (discussed in subsection 3.1.5).
The non-dimensionalized equations are (note abuse of notation: tilde’s denoting
non-dimensionalization have been dropped):
42
τ = A
∫∞
−∞
sx(x− ξ)
πξdξ + Bsx (3.13)
qs = 1 +3
2τ (3.14)
qx =
⎧⎪⎨⎪⎩qs − q h > ε and h = s
0 h < ε and h = s
−q h < s.
(3.15)
ht = −qx + (D(hx)hx)x (3.16)
3.1.8. Simulation algorithm
The simulation evolves the non-dimensional dune profile h over a time step dt, via
several steps:
1. Consider the dune profile h(x, tn);
2. Apply diffusion to the surface, causing “instantaneous” avalanches where the
slope exceeds the angle of repose (this is done first to eliminate unrealistic steep
slopes);
3. Calculate separation bubble s(x);
4. Calculate shear stress perturbation τ(s);
5. Calculate saturated sand flux qs(τ );
6. Calculate actual sand flux via qx(qs) equation;
Note that it is this step which preserves nonnegativity of h, as qx ≤ 0 (only
deposition can occur) as long as h is small (i.e., can’t erode below the “bedrock”
surface: h < ε).
7. Calculate evolution in dh(qx);
8. Iterate for next time step: h(x, tn+1) = h(x, tn) + dh.
43
Further details of the numerical algorithm are given in Appendix A.
3.2. Model Analysis
Studies of isolated dune evolution are able to replicate reasonable dune profiles of
many sizes and types, such as tranverse dunes (Figure 3.3) or three-dimensional
barchans (e.g., Hersen 2004). Comparison between simulated dune forms and be-
haviors and observed dunes is the primary method used for validation of the model
equations. A more quantitative exploration of observed relations is also done through
numerical simulation and analysis.
−50 −40 −30 −20 −10 0 10 200
2
4
6
8
10
12
14
x [m]
h [m
]
Figure 3.3. Sample steady-state two-dimensional terrestrial aeolian dune profilescalculated through numerical simulation. Different dune cross-sectional areas areshown, from a roughly minimum-sized transverse dune (dunes with height less than1m consist of a convex windward slope with no slipface; these are known as domedunes) to a dune 14m in height. Note that the vertical axis is exaggerated – theaverage windward slope for these profiles is ∼ 10o, which is comparable with mea-surements (Cook et al. 1993, Parteli, Schwammle, Herrmann, Monteiro & Maia 2005).
3.2.1. Linear stability analysis
Linear stability analysis of similar model equations (without the diffusion avalanche
term) about a perfectly flat plane with flux at the saturated flux value (h(x, t) = H0
(> ε) and q(x, t) = 1) was originally done by Andreotti et al. (2002b). We simply
44
restate the results here, as we assume the initial pertubation is sufficiently small that
diffusion can be neglected (i.e., D remains small). For perturbations of the form:
h = H0 + H exp(σt + ikx) (3.17)
q = 1 + Q exp(σt + ikx) (3.18)
we have a perturbation growth rate of:
σ =3k2(−A|k|+ B)
2(1 + k2)(3.19)
Thus, the instability grows in time when
B > A|k| ⇒ |k| < B/A; (3.20)
i.e., we expect a long wavelength instability. Taking just the linear portions of
dσ/dk = 0 we see a maximal growth wavenumber:
kmax =2B
3A(3.21)
2π/k = λmax =3πA
B(3.22)
Plugging in A = 4, B = 1, we have a wavelength of∼ 38, which is observed numerically
to be the first wavelength of growth.
Redimensionalizing with ls ∼ 1m for terrestrial dunes or ls ∼ 13m for martian
dunes (Claudin & Andreotti 2006), we approximately replicate the mean dune sizes
observed on these two planets (Figure 3.4) of around 30m and 500m, respectively.
This demonstrates that the model equations used do capture the initial dynamics
of dune growth, which depends on the characteristic length scale ls. Further dune
growth is due to nonlinear effects, such as flow separation and dune interactions.
3.2.2. Reduced dimensional analysis
To further simplify the dune model, we neglect the detailed dune morphology and
instead approximate the dune form with a piecewise linear shape (Figure 3.5). This
45
Figure 3.4. Histograms of the dune wavelengths measured on (a) Earth and (b)Mars. (a) The solid line shows barchan dune wavelengths throughout a Moroccondune field, and the dashed line shows dune wavelengths on the windward side of amega-barchan within that field. (b) The solid line shows dune wavelengths measuredwithin in several intracrater fields, and the dashed line considers only Kaiser crater(19E, 47S) dunes. Averaged wavelength values are 20 m (solid line on panel a), 28 m(dash line on panel a), 510 m (solid line on panel b) and 606 m (dash line on panelb). Images taken from Claudin & Andreotti (2006).
46
allows us to describe the profile with just two variables: m(t), the horizontal location
of the dune brink, and σ(t), the stoss slope:
h(σ(t), m(t)) =
⎧⎪⎨⎪⎩σ(t)(x−m(t)) + H, m(t)−H/σ(t) < x < m(t)
- tan(34o)(x−m) + H, m(t) < x < m(t) + H/ tan(34o)
0, otherwise.
(3.23)
The dune height H can be determined as a function of σ if the volume V (which
is the area of the dune shape in one dimension) is specified and held constant:
H(σ) =
√2V
1/σ + 1/ tan(34o). (3.24)
34
Figure 3.5. This is the simplest representation of a dune profile, where given aninitial and constant dune area (V), only two variables (m and σ) are needed to definethe dune shape and location.
We consider temporal evolution of m and σ by projecting the right hand side of
Equation (3.6) (dh/dt = (∂h/∂m)(dm/dt) + (∂h/∂σ)(dσ/dt) = -qx) onto the space
spanned by ∂h/∂m and ∂h/∂σ. To complete this projection, we use the inner product:
〈f, g〉 =
∫ m(t)
m(t)−H/σ(t)
f(x)g(x)dx (3.25)
and the functions v1 = 1 and v2 = x −m + L/2, which span the same subspace as
∂h/∂m and ∂h/∂σ over the dune’s stoss slope.
The second inner product results in:
〈−qx, v2〉 =L2
12
dσ
dt(3.26)
47
This equation gives the evolution of the windward side angle which is independent
of m. With this analysis, we can see that for a given dune mass, the stoss slope (σ)
will approach an equilibrium value, implying the existence of an equilibrium dune
shape for a given dune size. However, this slope is higher than observed transverse
dune stoss slopes, which have average values of ∼ 10o (Momiji et al. 2000, Parteli,
Schwammle, Herrmann, Monteiro & Maia 2005) and peak values of ∼ 15o (Cooke
et al. 1993). Thus, detailed dune morphology and dynamics must play a role in
setting observed dune shapes and sizes.
5 10 15 20 25 30−5
−4
−3
−2
−1
0
1
2
3
4
5x 10
−3
σ (windward slope)
dσ/d
t
V=50
V=200
V=800
Figure 3.6. Dynamics of windward slope as a function of slope for several differentsize dunes (the values V refer to the dune’s cross-sectional area). For each dune size,there is a single stable equillibrium value (for large dunes, this is about 18o).
The first inner product yields:
dm
dt= − 1
H
[〈−qx, v1〉 − dσ
dt(L
∂H
∂σ− L2/2)
]. (3.27)
If the dune has a steady shape (dσ/dt = 0), then the dune migration velocity will be:
dm
dt= − 1
H〈−qx, v1〉 =
q(m−H/σ)− q(m)
H. (3.28)
48
and we reproduce the observed inverse relationship between the dune height and its
velocity (Bagnold 1941).
A reduced-dimension model was also developed for a two-dune system, and is
described in Appendix B. This model was used to investigate binary dune collisions,
but the large number of coupled variables needed to describe the system (12 versus 2
in the one-dune system) did not simplify the simulations or analysis, so this approach
was abandoned.
49
4. INFLUENCE OF SAND FLUX
As the dune evolution model was constructed based on the averaged-dynamics of
sand transport, we begin our analysis by considering the influence of sand flux on an
individual dune’s evolution. In this chapter, we then extend those results to evaluate
dune evolution within a field, where dunes interact through sand flux (i.e., a dune’s
influx sand amount is set by the upwind dune’s outflux).
4.1. Physical Description
As discussed in Section 2.1, sediment supply and sand flux dynamics exert crucial
influences on dune evolution. The sand flux over a dune is one of the main measure-
ments made within field studies of dunes. However, estimates of the total saltating
sand flux cannot be easily related to dune migration, as:
• dunes are not perfect collectors. A large fraction of the incoming sand flux can
be lost off the horns of barchan dunes (Hersen 2004), and sand can also be lost
from transverse dunes (Momiji & Warren 2000);
• dune migration and growth is related to the net (not gross) sand flux over a
dune;
• the sediment flux available to a field generally originates at a point or line
source(s), and thus is not uniform within a field. Instead, the sediment flux
available to a dune within the field depends primarily on the amount of sand
escaping from the nearest upwind dune(s) and topography.
Thus, to consider dune evolution within a field, it is necessary to consider the exchange
of sand between dunes.
50
4.2. Model Results
As the goal of these simulations is to evaluate the effect of sand flux (not total sand
available), simulations were run with a semi-infinite boundary condition with fixed
influx.
4.2.1. The need for a non-zero influx
Laboratory (Katsuki, Kikuchi & Endo 2005) and field observations (Hersen 2004)
show that dunes will lose sand downwind through crest defects and other three-
dimensional features, such as barchan dune horns, as they evolve and migrate. This
loss is enhanced through stochastic variations in wind direction and strength. Thus,
an incoming sand flux is necessary for a dune’s continued existence.
In our simulations, these three-dimensional effects are neglected; we essentially
consider only defect-free transverse dunes, so care must be taken in comparing our
two-dimensional simulation results to three-dimensional dunes. When the dune form
is constrained to two dimensions, the slipface can act as a perfect collector, losing
sand only when the dune slipface length is small (so that some saltating grains pass
completely over the lee slope). Measurements of sand flux over the brink of a ter-
restrial desert transverse dune showed that 55%–95% of the sand flux is deposited
within 1m of the crest, and 84–99% within 2m (Nickling et al. 2004).
From equation 3.5, we can see that sand flux decays exponentially once past the
dune brink from equation 3.5 (and qs = 0 in the separation bubble):
qx = − 1
lsq (4.1)
⇒ q(x) ∼ e−1
lsx. (4.2)
As the lengthscale for sand flux decay is ls, we have a minimum slipface height that
will scale with ls tan(34o) (as the slipface is at the angle of repose). For Earth,
the minimum dune height at which a slipface develops and is stable is 1m, which
51
is about twice the scale height. As a dune’s height increases beyond this amount
(H ls tan(34o)), it should very efficiently collect sand.
Observations and modeling studies have found that the sand-trapping efficiency of
the slipface increases as the dune grows (Momiji & Warren 2000) even within three-
dimensional dunes. This creates an unstable equilibrium dune size (at which the sand
influx equals the sand outflux). Smaller dunes will experience an net loss of sand and
shrink in size, while larger dunes will experience a net gain and grow.
4.2.2. Sand flux over a dune
In previous studies of dune evolution, there has been scant attention paid to the influx
sand amount beyond a general assumption that it is small with respect to the far-
field saturated sand flux (qin = Cqs,∞, C ∼ 0.2). Observations of the interdune sand
flux generally support this assumption (Fryberger et al. 1984); it is thought that the
interdune sand flux remains limited to a small fraction of the saturated sand flux due
to the fluctuations in wind speed enhancing deposition onto small-scale topography
when a larger amount of sand is mobilized. Additionally, any sand available for
saltation within interdune regions is rapidly eroded as the winds downwind of dunes
and the separation bubble accelerate as they recover, and are typically undersaturated
in sand flux (Fryberger et al. 1984, Baddock et al. 2007).
Despite this apparent natural limitation in keeping the influx sand amount (equiv-
alently, the multiplicative factor C ∈ [0, 1]) small, we investigate the influence this
free parameter exerts on dune evolution model.
Simulations of a dune moving over flat topography showed two types of behaviors:
for low C values the dune migrated, and for large values C the dune did not migrate
but instead grew in height and length with an anchored foot. (For very large values
of C, the dune also grew backwards due to the slight decrease in shear stress at the
toe of the dune which caused deposition.) These different behaviors were caused by
52
differences in the amount of sand eroded from the foot and stoss slope of the dune,
to be deposited on the lee side.
Simple physical scaling arguments yield the threshold between different values.
The total amount of additional sand that the wind can carry when it encounters the
dune is (1 − C)Q, while the amount of sand that must be moved for the dune to
migrate is the mass of the dune (M). Thus, the threshold between dune migration
vs. dune growth depends on (1 − C)Q/M (which has units of inverse time): when
(1 − C)Q/M is smaller than a threshold amount the dune will remain in place and
grow, and when (1− C)Q/M is larger, the dune will migrate (Figure 4.1).
50 100 150 200 250 300 350 4000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
dune cross−sectional area (M)
fract
ion
of s
atur
ated
san
d flu
x er
oded
(1−C
)
Figure 4.1. Plot showing that different dune dynamics: migration (*) or stationarygrowth (�), occur depending on the sand influx amount relative to the dune size.The boundary between the two dynamics is found when (1 − C)/M ∼ 0.002; withthe far-field influx (let Q ∼ qs,∞): (1− C)Q/M ∼ 5e-6.
Thus, we again see unstable dynamics – for a given C and Q, large dunes will
grow while smaller dunes migrate.
53
4.3. Physical Implications
Our results provide a negative conclusion with regards to controls on maximum trans-
verse dune size: sand flux effects alone cannot regulate dune size within a field.
For a given sand flux, simulated single dunes instead grow without bound (limited
only by the total amount of sand available for accumulation). Previous continuum
(Schwammle & Herrmann 2004) and discrete (Momiji 2001) dune model studies found
that the average transverse dune height within a field scales with the square root of
time.
Combining this result with a line source of sediment for the field, and the increase
in sand-trapping efficiency as a dunes grows means that dunes within a field should
exhibit coalescent behavior: larger dunes will collect all upwind sand, while smaller
dunes will become sand-starved and shrink in size, causing them to release more sand
to the larger dunes (through increased downwind sand flux and collisions).
This analysis and our results are comparable to a study of barchan dune evolution
by Hersen et al. (2004) (Fig. 4.2). That study found that solitary barchans and
barchan fields are unstable in the case of a constant wind, with instabilities developing
within a field of 100m wide dunes after a century, over a migration distance of a
few kms. As observed dune fields hundreds of kilometers long do contain dunes of
uniform size, Hersen et al. (2004) concluded that there must exist another dynamical
mechanism that stabilizes the dunes sizes.
We concur with this conclusion, and seek to identify a process(es) that limits
dune size within a field, or that manages to balance the flux instabilities (at least
within larger dunes). In the following chapters, we investigate the influence that
dune interactions and bedrock topography can play in dune and dune field evolution.
54
Figure 4.2. Sketch illustrating dune size instability within a field due to the ex-change of sand between the dunes: small barchan dunes experience a net loss ofsand, which causes their downwind neighbors to grow. This process continues asthe small dunes shrink and the large dunes grow, causing coalescence as the sand isconcentrated in a smaller number of larger dunes. Image is from Hersen et al. (2004).
55
5. INFLUENCE OF DUNE INTERACTIONS
Studies which consider the evolution and behavior of a single dune cannot be directly
extended to evolution of dunes within a field, as isolated dunes do not behave and
evolve the same way as dunes within a field. For example, as discussed in Chapter 4,
individual dunes are unstable with respect to changes in the sand flux. This implies
that dunes in a field, evolving only through sand flux, should eventually coalesce into
a small number of large dunes (Hersen et al. 2004); as this does not occur, other
processes must aid in limiting the sizes of dunes within a field.
In this chapter, we investigate the effect of neighboring dunes on shear stress and
saltation. We also consider the dynamics of binary dune collisions, and evaluate evo-
lution within a field through binary dune collisions. We find that collision dynamics
can yield a field of similarly-sized dunes if the influx dunes’ sizes are chosen from a
sufficiently constrained population.
5.1. Physical Description
Wind tunnel (Walker & Nickling 2003) and field studies (Lancaster 1985, Baddock
et al. 2007) of isolated or closely spaced dunes have shown that the presence of non-flat
topography (such as terrain or neighboring dunes) will alter the wind-exerted surface
shear stress, which in turn changes sand flux and dune evolution. For example,
the distance before flow reattachment occurs is foreshortened by the presence of a
downwind dune, causing the flow velocity at the foot of this dune to be lower (Walker
& Nickling 2003). Wind speed up over the stoss slope is then found to increase faster
over paired dunes than over isolated dunes (Lancaster 1985) – potentially steadying
sand flux rates in the upper-portions of the stoss slope and increasing crestal erosion
(Walker & Nickling 2003).
56
5.1.1. Effects not included in this study
These changes in flow dynamics are caused by greater turbulence as the reattaching
flow encounters the downwind dune’s stoss slope and is compressed (Sweet & Kocurek
1990, Baddock et al. 2007). This added turbulence is not accounted for in the airflow
model used (subsections 3.1.1-3.1.2), so is not presently included in the dune evolution
model used in this study. Empirical studies are needed to provide constraints on how
the added turbulence will change sand transport and dune interactions; work by
Palmer (2010) and Ewing & Kocurek (2010) are the only available studies that have
begun to quantitatively examine this problem.
The interdune distance also is controlled by the reattachment and recovery of air-
flow in the lee of the upwind dune (Walker & Nickling 2002, Walker & Nickling 2003,
Schatz & Herrmann 2006, Baddock et al. 2007) (as described in subsection 2.2.4). This
spacing may be influenced by three-dimensional or time-varying factors, such as com-
plex secondary flow patterns (Walker & Nickling 2003), the incidence angle between
the wind and dune crest, or changes in atmospheric stability (Sweet & Kocurek 1990);
as the model used is two-dimensional and assumes steady airflow, these factors are
not considered. Thus, this study does not focus on the specific controls and pro-
cesses that control dune spacing, only dune sizes. Our method of modeling saltation
over sand-free interdunal areas (described in subsection 3.1.4) causes spacing between
simulated dunes to be proportional to the flow reattachment distance. This distance
we assume to be proportional to the upwind dune height (subsection 3.1.1), causing
our modeled interdune spacing to be related to dune sizes; this result matches to
first-order observations of barchan and transverse dunes (Lancaster 1988).
5.1.2. First-order effect of topography on shear stress
We first investigate the distance over which non-flat topography will interact with a
dune by considering the asymptotic behavior of the shear stress perturbation exerted
57
at a point x by a nearby dune:
Δτ(x) =
∫ dune slipface
dune foot
hx(x− ξ)
πξdξ (5.1)
If the dune is downwind of x, then the ξ denoting where the dune is located (hx(x−ξ) �= 0) will be negative. The side of the second dune closer (and thus of more
influence) is the stoss (positively sloped) side, so the overall effect of the shear stress
perturbation is negative. Similarly, if the second dune is upwind, the relevant ξ will
be positive and the more influential side of the second dune is the negatively sloped
side, so the overall effect on the shear stress perturbation is negative. This makes
physical sense, as additional topography would shift the wind-streamlines upward,
thus decreasing the shear stress exerted on a dune surface.
�������������
�
� �
Figure 5.1. Schematic diagram illustrating variables used in first-order topography-induced shear stress estimation.
To estimate the effect more quantitatively, assume a distance of d between the
point of calculation and the dune foot. The dune is of length l, so xdune foot = x + d,
xdune slipface = x + d + l (Figure 5.1). Translating the horizontal coordinate system to:
η = −ξ − d and assuming l � d yields:
Δτ =
∫ l
0
hx(x + d + η)
π(d + η)dη =
∫ l
0
1
πd
hx(x + d + η)
1 + η/ddη (5.2)
≈ 1
πd
∫ l
0
hx(x + d + η)(1− η/d)dη, (5.3)
58
neglecting all higher order terms. (The last expression derived is the first-order taylor
expansion of 11+x
where x� 1, as η/d� 1 within the interval of integration). As the
dune is migrating over a flat plane,∫ l
0hx(x+d+η)dη = 0. We integrate-by-parts the
remaining integral (with boundary conditions h = 0):
Δτ ∼ -1
πd2
∫ l
0
h(x + d + η)dη (5.4)
∼ − A
πd2(5.5)
∼ −( d
H
)−2
(5.6)
where A is the total dune cross-sectional area (A =∫ l
0h(x + d + η)dη) and H is the
dune height (assuming A ∼ H2; Figure 5.2).
101 102
10−3
10−2
10−1
100
d/H
|Δτ|
Figure 5.2. Log-log plot of the magnitude of the calculated Δτ as a function ofdistance from an upwind dune’s foot, normalized by the dune’s height (H ∼ 1
10l).
The different curves were computed for dunes of different sizes, with the expected -2-power (trendline) appearing when d > 30H (∼ 3l): Δτ ∼ (d/H)−2. The Δτ curvescurl up on the right as d becomes comparable to the computational domain size.
This topography-induced decrease in shear stress results in lower saltation rates.
As sand transport occurs at a slower rate, dune evolution, migration, and interactions
will also occur at slower rates. Although this does not put a limit on dune size, this
59
effect will play a (small) role in slowing dune growth as dunes within the field increase
in size. Additionally, this will provide more time for other stabilizing processes to
influence dune and dune field evolution.
5.2. Study of Dune Collisions
We investigate the dynamics of binary dune collisions by initializing our continuum
dune model with two dunes of specified size, the smaller located upwind of the larger.
The initial dune profiles were steady-state profiles calculated during simulations of
isolated dune formation (found with periodic boundary conditions). The initial dune
spacing was chosen by testing for the minimal distance at which no changes in size
occurred immediately in the downwind dune because of the presence of the upwind
dune (because of changes in its shear stress and, thus, sand flux calculation). To
eliminate a dependence on the sand influx rate (which influences dune migration and
growth, as discussed in subsection 4.2.2), periodic boundary conditions were used.
In these simulations, the upwind (and smaller, thus faster) dune catches up to the
downwind dune. As the shadow zone of the upwind dune impinges upon the foot of
the downwind dune, sand in the foot of the downwind dune is not transported by
the wind and so is left behind and is eventually absorbed by the upwind dune. As
the downwind dune loses sand, its profile gradually becomes shorter, which causes
its velocity to increase. Similarly, the upwind dune gains sand, becomes taller, and
decreases its velocity. Eventually, the upwind dune begins to climb the downwind
dune and the two dunes would lose their distinct shapes and form an amorphous
two-humped dune-complex. Sand continues to be exchanged and the humps change
in height (the upwind hump grows and the downwind hump shrinks) while moving
toward, then (once the downwind hump is the smaller of the two) away from each
other (Figure 5.3).
Simulations were run with combinations of dunes with cross-sectional areas of 10−
60
400m2 (corresponding to initial heights of 1− 15m), and dune pairs were categorized
by the resultant type of interaction (Figure 5.4).
We noted two qualitatively distinct outcomes: (1) coalescence resulted when the
downwind hump subsided into the dune complex, resulting in one unified dune, and
(2) ejection occurred when the downwind hump managed to separate completely,
resulting in an exchange of material between the colliding dunes. In a small number
of cases, the downwind hump managed to separate but was too small to remain stable
and so would eventually disappear. We initially examined this case separately; as it
occurred in only a small number of cases, we reclassified it as an extreme case of
ejection.
Plotting the different types of dune interactions as a function of the sizes of the
colliding dunes (downwind cross-sectional area: Abefore vs. upwind cross-sectional
area: abefore < Abefore, Figure 5.4), it can be seen that the boundary between coa-
lescence and ejection cases (i.e., the intermediary disappear cases) roughly follows a
straight line passing through the origin. Along that line the size ratio between the
two dunes before interaction is constant: r = (a/A)before ≈ 1/3. From this, we can
see that if we consider a pair of dunes such that the line from that point (A, a) to the
origin is less steep than that boundary (equivalently, the size ratio of the two dunes
is between 0 and 1/3), coalescence occurs. Conversely, when the line between a point
and the origin is steeper than the boundary (or the size ratio is between 1/3 and 1),
the collision results in two dunes or ejection. Thus, it is appears that the size ratio
(r) between the dunes is what determines a collision result, not the individual sizes
of the dunes.
5.2.1. Defining the interaction function
To understand in more detail what will result when two dunes collide, we also com-
pute the size ratio of the dunes after collision (f(r) = (adownwind/Aupwind)after; where
61
coalescence ejection
wind
2
1
3
4
5
Figure 5.3. Schematic diagram illustrating a dune collision. The topography/duneprofiles are outlined in black, and the separation bubble is outlined in gray. Thehorizontal arrows show relative dune velocity and the vertical arrows show changesin dune height. From top to bottom: 1. The upwind (smaller) dune approachesthe downwind dune. 2. Eventually its separation bubble impinges upon the foot ofthe downwind dune, arresting that sand. 3. As the upwind dune continues to movetoward the downwind dune, it gains sand that the downwind dune leaves behind. 4.As the upwind dune grows and the downwind dune shrinks, their relative crest heightreverses, allowing the downwind dune to migrate faster than the upwind dune. 5.If the downwind dune loses all of its sand before it can migrate away, then we havecoalescence. Otherwise, we have an ejection case.
62
50 100 150 200 250 300 350 400
20
40
60
80
100
120
140
160
Adownwind [m2]
a upw
ind [m
2 ]
coalescencedisappearejectionslope = 1/3
Figure 5.4. Plot of the different types of dune interactions as a function of the sizesof the dunes before collision: the area of the downwind dune (A) vs. the area of theupwind/smaller dune (a). Note that the boundary between coalescence and ejectionroughly follows a straight line through the origin with slope of roughly 1/3. Thisimplies that if the dune size ratio before collision: (a/A)before is smaller than 1/3,then coalescence occurs. Conversely, if (a/A)before > 1/3, then two dunes result.
adownwind, after = 0 if the dunes coalesce), and plotted this against the size ratio before
dune collision (r). Resultant dunes which disappear are treated as very small ejection
dunes because the sizes are measured as soon as the dune-complex separates into two
dunes. As can be seen in Figure 5.5, the data collapses onto a single curve, indicating
that dune size ratio resulting from a collision is uniquely determined by the size ratio
before collision. We interpolate between the calculated points to generate the interac-
tion function, which we will use in the dune field model to relate the before-collision
size ratio (r) to the after-collision ratio (f(r)). We note that the interaction function
has the following characteristics:
• The function f(r) monotonically increases.
• When the initial size ratio of the colliding dunes is smaller than a threshold
amount (here r ≤ 1/3), f(r) = 0 indicating coalescence.
• In the limit where very similarly sized dunes collide, f(r) appears to → 1.
63
Similar results have been found elsewhere in the literature. For example, stud-
ies of discrete numerical simulations of interacting three-dimensional barchan dunes
(Katsuki, Nishimori, Endo & Taniguchi 2005) and laboratory experiments of subaque-
ous barchan dune collisions (Endo, Taniguchi & Katsuki 2004) yield results which
support our hypothesized interaction function characteristics. Additionally, a study
by Duran et al. (2005) using continuum numerical simulations of three-dimensional
barchan dune interactions found an interaction function (Figure 5.6) which also con-
sists of a continuous, monotonic relation between size ratios before and after collision,
with f(r)→ 0 as r → 0 and f(r)→ 1 as r → 1.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
r = (aupwind/Adownwind)initial
f(r) =
(ado
wnw
ind/A
upw
ind) af
ter coalescence (122)
disappear (16)ejection (93)
f(r) > 0
r = 1/3
f(r) = 0
Figure 5.5. Plot of the area ratio of colliding dunes, before collision vs. after.Note the distinct zones of interaction results: coalescence occurs when the dune sizeratio before interaction is below some threshold (1/3); when the ratio is above thatthreshold, then the output size ratio generally falls along a specific curve, independentof the absolute sizes of the dunes (the outlier corresponds to the smallest dune pairsampled). In the legend, the numbers given are the total number of simulations whichyielded that particular type of interaction.
64
Figure 5.6. This plot was derived for three-dimensional barchan dune collisions byDuran et al. (2005) and shows volume ratios of colliding barchan dunes before versusafter collision (the dots, along with a best fit curve). It exhibits the same characteris-tics as the interaction function shown in Figure 5.5: a threshold between coalescence(c) versus all ejection-type interactions (b/bu/s: breeding, budding, solitary waves),a single monotonic curve along which all points fall and which passes through (0,0)and probably (1,1). The straight line shown has a slope of one.
65
5.2.2. Dependencies of the interaction function
The balance between the timescale over which the upwind dune merges with the
downwind dune and the rate at which the downwind dune shrinks (and migrates
faster) is what determines the end result of a collision.
For example, the timescale over which the dunes merge is very short if dunes
initially have very different sizes, as the relative velocity between the dunes is large.
The dunes will merge together before the downwind dune can shrink sufficiently to
escape, which is why f(r)→ 0 as r → 0. Conversely, the rate at which the downwind
dune shrinks is more important if dunes are initially very similar in size, because
the relative velocity will be low. As they will move towards each other over a long
time period, the downwind dune will be able to slowly escape after becoming slightly
smaller than the upwind dune. The two dunes will end up close to each other in size,
so f(r)→ 1 as r → 1.
However, the range of the interaction function does not include the end-point (1)
as it is not possible for an interaction to yield two exactly similar dunes – these two
dunes will move at exactly the same velocity, whereas an ejected dune must be able to
move away from the upwind dune. By similar reasoning, the domain of the function
also does not include 1 as it is not possible for dunes migrating at the same velocity
of collide. The range (but not the domain) does include 0 as long as coalescence can
occur.
A continuum between these two effects is expected, as the timescale over which
dunes interact will increase as the disparity between the sizes of the two dunes in-
creases. This yields a continuous, monotonic function.
In summary, we expect an interaction function to have the following characteris-
tics:
• f : (0, 1)→ [0, 1),
66
• continuous and monotonically increasing,
• f(r)→ 0 as r → 0,
• f(r)→ 1 as r → 1.
The exact form of the function will depend on factors which set the timescale over
which the dunes interact and the rate of the sand exchange between the dunes. In
the continuum dune model, for example, one way to change the rate at which sand
is exchanged is by increasing the aspect ratio of the separation bubble (β). This
lengthens the shadow zone of the upwind dune, causing the foot of the downwind
dune to be caught in that shadow zone and arrested sooner. The downwind dune,
thus, loses more sand before the upwind dune begins to gain sand. As the growth
of the upwind dune is delayed relative to the shrinkage of the downwind dune, the
downwind dune will need to shrink more to become the smaller of the two. When
the downwind dune is finally ejected, it will be smaller (and the upwind dune will be
larger), causing the interaction function to be lower (i.e., f(r) will decrease ∀r where
f(r) > 0). Conversely, if the length of the separation bubble is decreased, then the
interaction function will be higher. Test simulations have shown this to be the case,
although the effect was small (Figure 5.7).
5.2.3. Implications of the crossover value
Our interpolated interaction function for transverse dunes (Figure 5.5) is included in
this study to explain the general form of an interaction function. The specific function
that was found through our two-dimensional dune simulations, however, will not be
used in the remainder of this study because that functional form (where ∀r, f(r) < r)
yields coalescence-dominated dynamics.
When ∀r ∈ (0, 1), f(r) < r, then in every collision the larger dune grows and the
smaller dune shrinks. As the smaller dune in every collision shrinks and eventually
67
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
r = (aupwind/Adownwind)before
f(r) =
(ado
wnw
ind/A
upw
ind) af
ter
Figure 5.7. This plot shows the best-fit interaction functions for transverse dunes,after using different separation bubble aspect ratios (β) in the continuum dune model:from top, β = 3, 6.5 (shown in Figure 5.5), 8, 10. As hypothesized, f(r) decreases asthe separation bubble aspect ratio is increased (lengthening the length of the shadowzone) due to its influence on how quickly the downwind dune is able to lose sand.
coalesces with the larger dunes (for small enough r, f(r) = 0), the dune field perpet-
ually evolves into a system containing a smaller number of larger dunes. This type
of dune field dynamics will never yield a stable pattern of similarly sized dunes, and
will in fact never be stable (as shown in Proposition 1).
Proposition 1 (Guaranteed Coalescence). Let us consider a dune field with a con-
tinuous influx of dunes chosen from a fixed size population with some variance (i.e.,
not a delta-function). If this field’s interaction function f satisfies the following:
1. ∀r ∈ (0, 1), f(r) < r, and
2. ∃rc such that ∀r < rc, f(r) = 0,
then the dune field will perpetually exhibit coalescent dynamics.
Proof. If the influx dune size population is fixed, then the average size of a dune
within the field (M =∫∞
0Mp(M)dM , where p(M) is the probability density of dune
68
sizes within the field) will be the same as the average dune size within the influx
population unless coalescence occurs and the number of dunes decreases. Thus, it
is sufficient to show that interactions resulting in coalescence will continuously occur
within the field.
This can be seen by considering the variance of p(M) (given by the second moment:∫∞
0M2p(M)dM). As ∀r, f(r) < r, then mass is distributed from the smaller to the
larger dune (m → m− ε and M → M + ε, ε > 0), and:
(m− ε)2 + (M + ε)2 = m2 + M2 + 2(M −m)ε + 2ε2
> m2 + M2. (5.7)
Thus, we see that the variance will increase with every binary dune collision. As the
mean is unchanging (unless coalescence is occurring), then this implies that, after
a sufficient number of interactions, small dunes will be created that will experience
collisions with r sufficiently small (r < rc) that f(r) = 0. This will only be negated
if the dune field becomes size-sorted (i.e., all of the small dunes are at the front
of the field), but size-sorting cannot occur as long as the (fixed) influx of dunes is
continued.
Conversely, when ∀r, f(r) > r, then in all collisions the dunes become more similar
in size. Eventually, after many collisions, the dune field will always evolve into a
system of many similarly-sized dunes (with any remaining collisions occurring around
r ∼ 1), as shown with Proposition 2.
Proposition 2 (Guaranteed Patterning). Let us consider a dune field with a con-
tinuous influx of dunes chosen from a fixed size population. If this field’s interaction
function f is such that ∀r ∈ (0, 1), f(r) > r. then the dune field will evolve into a
system of similarly-sized dunes.
Proof. We again consider the the variance of the probability density of dune sizes
within the field (∫∞
0M2p(M)dM). As ∀r, f(r) > r, then mass is distributed from
69
the larger to the smaller dune (m → m + ε and M → M − ε, ε > 0), and:
(m + ε)2 + (M − ε)2 = m2 + M2 − 2(M −m)ε + 2ε2.
This is smaller than m2 + M2 only if ε < (M − m), but this is within the defined
limit of mass exchange (ε < (M −m)/2)as the small dune cannot become larger than
the large dune (or it would be the large dune). Thus, we see that the variance will
decrease with every binary dune collision, implying that the system moves towards a
population of similarly-sized dunes.
In most systems, the variance will be kept non-zero through size-sorting (as small
dunes escape at the front of the field) and as the influx is continued (if coalescence
has occurred, M within the field will be larger than the mean size within the influx
population).
As one more extreme example: if ∀r, f(r) = r, then dunes will behave similar to
solitons in that all interactions will preserve the sizes of the dunes involved, as if the
dunes simply pass through each other.
Most natural dune field interaction functions will not correspond with one of
these examples (in particular, see Livingstone et al. (2005) for a discussion about the
unphysical nature of the soliton example). Instead, a natural dune field interaction
function will probably be a combination of these extreme examples, with some regions
where f(r) < r, some regions where f(r) > r, and transition points where f(r) = r. In
these cases, the dynamics of the dune field can be determined based on the interaction
function crossover value: the value r∗ such that for all higher r < 1, f(r) > r. We
define the crossover value as the lower bound on the region connecting to (1,1) where
f(r) > r. An interaction function may have several regions where f(r) ≥ r, but only
the region including (1,1) is of interest, as it is necessary for f(r) > r as r → 1 for
interactions to actually push the system toward a field where all dunes are about
the same size. Additional lower regions where f(r) > r will affect the timescale over
70
which the system evolves, but not the end state. Even in simulations with a carefully
paired influx dune population and interaction function (such as a strongly-peaked
bimodal dune size distribution, with f(r) = r at the ratio of the two peaks and
r∗ = 1), eventually a collision results in one dune of sufficiently different size that the
field is pushed out of a patterned formation.
To further see that r∗, as defined, is the important parameter, we approximate the
interaction function as a map from interval [0, 1] onto itself. In a true discrete-time
dynamical system where f(r) is iterated over the unit interval, we have an invariant
set (r∗, 1) which is bounded by an unstable equilibrium point (r∗) and the stable
equilibrium point (1) (i.e., applying fn(r)→ 1 as n →∞, if r ∈ (r∗, 1]. Although this
is not a perfect approximation to the interaction function (as dunes do not repeatedly
collide with each other: although a given collision may have r > r∗, subsequent
collisions involving those dunes may involve other dunes that are sufficiently different
in size to yield r < r∗), the argument still applies as a collision which has r > r∗ will
yield dunes which are each more likely to have subsequent collisions which involve
r > r∗ (as the variance of the system has decreased, as discussed in Proposition 2).
Thus, for a dune field model to possibly form a stable patterned system, its in-
teraction function needs to have r∗ < 1. Generic interaction functions with this
characteristic (in addition to those characteristics outlined in subsection 5.2.1) will
be considered in constructing the dune field model.
In Diniega et al. (2010), we showed that the precise details of the interaction
function are unimportant, and only the interaction function crossover value has a
significant influence on the end state of the simulation. Based on this, we use simple
piecewise-linear functions (Figure 5.8) for the remainder of this study, where the only
adjustable parameter is r∗.
71
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
r = (aupwind/Adownwind)before
f(r) =
(ado
wnw
ind/A
upw
ind) af
ter
0.8
0.1
0.5
Figure 5.8. Examples of the piecewise-linear interaction functions used in the re-mainder of this study; r∗ = 0.1 (thin), r∗ = 0.5 (medium), and r∗ = 0.8 (thick).Simulation tests showed the exact functional form is not important when consideringthe end state of the simulation – only the crossover value (r∗) matters. Thus, simplefunctions like these can be used as valid representatives of all reasonable interactionfunctions with the same r∗.
5.2.4. Other studies and their interaction functions
As we showed above, if r∗ = 0 (∀r, f(r) > r), the system will always achieve a
pattern of similarly sized dunes. Conversely, if r∗ = 1 (∀r, f(r) ≤ r), the system will
never have a stable patterned structure. This latter is the case we found with our
two-dimensional simulations of colliding dunes (Figure 5.5): ∀r, f(r) < r, so zero
was an attractive fixed point and the dune field moved toward perfect coalescence as
long as collisions occurred.
This was an odd and unexpected result, as a study by Pelletier (2009) had man-
aged to simulate the creation of patterned transverse dune fields (Figure 5.9) using a
discrete model. A key difference between our model equations concerned the calcula-
tion of the shear stress exerted by the wind. In Pelletier (2009), a nonlinear correction
was introduced to account for the added turbulence that develops as flow compresses
over steeper slopes, causing a decrease in shear stress: τ = τJH(1− wh2x), where τJH
is the shear stress calculated with the Jackson-Hunt model (Equations 3.1 and 3.2).
72
In adding this modified shear stress calculation into the binary collision simulations,
the interaction function did change in form (Figure 5.10) and was shifted upwards
as r → 1; however it still did not yield an r∗ < 1. Additionally, even the use of the
linear Jackson-Hunt shear stress calculation within the discrete model appeared to
generate stable dunes (Figure 5.9). Thus, other differences between the model used
in Pelletier (2009) and this study must account for the creation of the pattern, as
the influences of collisions and sand flux are not sufficient. Currently, no continuum
transverse dune model has resulted in a patterned dune field.
Figure 5.9. Plot of normalized bedform height (h/z0) versus time for a discretebedform evolution model using a linear (open circles) and nonlinear (close circles)shear-stress calculation (Pelletier 2009), based on an average of ten simulations. Withshort evolution times, height and spacing grow proportionately to the square root oftime (dashed line); over long-times, the dunes reach a steady-state condition in whichheight and spacing do not increase significantly with time. Figure is taken fromPelletier (2009).
The interaction function derived for collisions between barchan dunes in previous
studies (e.g., Duran et al. 2005) all have r∗ < 1 (r∗ ∼ 0.12 in Figure 5.6), which
accounts for why those studies created patterned dune fields. We extend the results
of those studies, in showing that when r∗ < 1, the interaction functions may yield a
patterned structure.
73
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r = (aupwind
/Adownwind
)before
f(r) =
(ado
wnw
ind/A
upw
ind) af
ter
Figure 5.10. Interaction function (large black markers) found with a modified shearstress calculation, which takes into account the decrease in shear stress over steepslopes, due to turbulence that develops as flow compresses (Pelletier 2009). Thisfunction steepens for higher r values, as compared to the function found with ourdune model (small gray marker), but still has r∗ = 1. Markers follow the conventionof Figure 5.5.
5.3. Field Model Description
As it is not reasonable to use the dune evolution model to examine the evolution and
interactions of the hundreds of dunes within a field, a multiscale approach is instead
used. Within the dune field model, the dunes themselves are treated as particles with
morphologies and dynamics approximated using simple phenomenological relations,
such as the interaction function.
5.3.1. Approximation of single dunes
Rather than keep track of every degree of freedom in the continuum model, it is useful
to only track dunes according to their size and location. To determine when dunes are
74
close enough to interact, we make reasonable assumptions about their morphology
and zone of influence. Dune shape is assumed to be scale-invariant, approximated as
a triangular wedge with a stoss aspect ratio of 10 (Parteli, Schwammle, Herrmann,
Monteiro & Maia 2005), a lee aspect ratio of 1.5 (angle of repose), and a separation
bubble with an aspect ratio of 6 (Schatz & Herrmann 2006) (Figure 5.11).
The dunes move with a velocity inverse to their crest height (Andreotti et al.
2002a); i.e., v ∼ k/H . The coefficient k varies between actual dune fields (Bagnold
1941). In this study, it is arbitrarily set at 100m2/yr.
Figure 5.11. Approximation of the morphology of a dune, with fixed stoss aspectratio of 10 and lee aspect ratio of 1.5. This shape is used to determine the dune’s zoneof influence as a function of area - two dunes interact when the separation bubble(with aspect ratio of 6) of the upwind dune touches the foot of the downwind dune,as shown.
5.3.2. Interactions and initialization: particle model
We initially attempted to use a Smoluchowski coagulation-type study to analyze dune
field evolution (described in Appendix C), as such equations are commonly used to
describe the evolution of large populations (by evolving the number density P of
particles of size x at a time t). Unfortunately, the dune field system sufficiently differed
from the physical systems considered by Smoluchowski equations (such as molecular
interactions within a vapor) that this approach became unwieldy: as sediment comes
from a point or line source, it made more sense to use a semi-infinite field instead of
75
periodic boundary conditions; secondly, dunes do not coagulate, but instead exchange
mass when they collide. Thus, this approach was abandoned in favor of a deterministic
particle model.
Our particle dunes move within a one-dimensional semi-infinite field. Collisions
occur when the foot of the downwind dune is touched by the separation bubble of the
upwind dune. The distance between dunes after collision is calculated the same way,
with the downwind dune located just outside the separation bubble of the upwind
dune.
The results of dune collisions are governed by the interaction function, which
relates the size ratio of the dunes before collision to the size ratio of the dunes after
collision (as described and derived in subsection 5.2.3). In our dune field model,
collisions occur instantaneously once dunes are close enough to interact.
Initial cross-sectional areas for the dunes (for either initial or influx dunes) are
taken from a specified range with uniform distribution. When we consider a semi-
infinite domain with an upwind influx of dunes, we assume a constant mass influx rate
of 30m2/yr to relate injection frequency to dune size. In general, this corresponded
to ∼ 1000 influx dunes per 1000 years for simulations run with a small mean dune
size (M = 40m2), and ∼ 400 when a larger mean dune size (M = 100m2) was used.
5.3.3. Model assumptions
Here we highlight and explain several assumptions we have made (in decreasing im-
portance) in designing our dune field model:
1. Sand flux effects are ignored. This means that dunes only change size when
they collide and the total size of colliding dunes is conserved during a collision
(i.e., (a + A)before = (a + A)after). In reality, sand can be lost or gained by
dunes between and during collisions (Elbelrhiti et al. 2008). Including this
sand flux, however, adds an additional layer of complexity on the model, and
76
such processes are not currently constrained by observations or experiments.
The effect this assumption may have on simulation results will be addressed in
subsection 5.4.3.
2. The interaction function is assumed to be spatially and temporally constant.
Dune collisions occurring at different times or locations are assumed to obey
the same interaction function. The interaction function, however, may depend
on local conditions, such as the type of sand included in the dunes. As timescale
is not considered, in this study it is not important that the interaction function
may change shape. If the interaction function crossover value should change
temporally or spatially, however, as is suggested in Besler (2002), then this
could impact the end state of the dune field. The physical implications of this
will be discussed in subsection 5.4.3.
3. Collisions happen instantaneously. We considered this to be an acceptable ap-
proximation as we observed that the time between collisions was much longer
than the collision timescale in our simulations. Additionally, this simplifies the
collision-dynamics as only two dunes can collide at a time.
4. Dune sizes are chosen from a uniform distribution. Although physical systems
generally have other types of distributions (e.g., Gaussian), a uniform dune size
distribution will be used in this study as its structure is the simplest. This will
remove one level of complexity from analysis of simulation results. Additionally,
we will discuss the effect of a Gaussian distribution of dune sizes in subsection
5.4.3.
5. The dune shape and zone of influence are specified somewhat arbitrarily. Both
of these will affect the timescale of dune field evolution, and the zone of influence
directly relates to the interdune spacing of a patterned system. These values will
77
have no effect, however, on whether or not a system will form stable similarly-
sized dunes, which is the focus of this study.
6. The value of the constant influx rate and the coefficient in the velocity relation
are also arbitrarily specified. Again, both of these constants will play a role in
the timescale over which the field attains its end state. As long as dunes are not
injected on top of one another, they have no other influence on the simulation.
5.3.4. Possible field end states
This study is concerned with the long-time behavior of dune fields – with this as-
sumption, a dune field can evolve towards two possible outcomes: (1) a dynamic
equilibrium is established where all dunes are and remain similar in size; we call this
end state quasi-steady (for example, the crescentic dunes in White Sands, New Mexico
or the martian dune field shown in Figure 1.2); or (2) the field may undergo runaway
growth, with one or a few dunes continually growing through coalescence with upwind
dunes (a possible example of this type of dune field structure on Mars is shown in
Figure 1.3).
Let us emphasize that because we are concerned with only the end state of a
simulation, the exact timescales for evolution are not considered in the following
tests. The effect timescale will have when model results are compared with actual
dune fields will be discussed in subsection 5.4.3.
In the quasi-steady end state, all dunes have approximately the same size and
interdune spacing – the epitome of a patterned dune field. The field will evolve into
the following characteristics:
• Since the dunes will become similar in size, they will move with approximately
the same velocity and, thus, rarely interact with each other.
• The interdune spacing will be similar across the field, as this quantity depends
78
on their defined zone of influence, which depends on their size (subsection 5.3.2).
• Even if new dunes with a range of sizes are introduced (e.g., through an influx
condition), interactions ultimately yield dunes of similar size with the rest of
the field.
• This end state is characterized by a linear increase in the number of dunes and
a steady mean dune size.
In the runaway growth case, one or a few dunes become large enough relative to
their surrounding dunes that all future interactions occur between very disparately
sized dunes (r small), which results in the larger dune getting still larger (and coalesc-
ing with all smaller upwind dunes). This will result in the following characteristics:
• The system will eventually settle into a size-sorted field, with dune size mono-
tonically decreasing with distance from the upwind boundary.
• Influx dunes will generally be smaller than the dunes found at or near the be-
ginning of the field, so will interact with a few dunes at the beginning, generally
losing sand and eventually coalescing with an anomalously large dune.
• This causes the number of dunes in the field to be quasi-constant.
• If periodic boundaries had been used, then no size-sorting can occur and the
final outcome is just a single large dune.
5.4. Results
Our results show that a dune field’s end state depends only on the interaction function
crossover value and influx dune size distribution’s standard deviation/mean ratio. The
way in which these two parameters are coupled can be seen most clearly in Figure
5.12, which shows a simulation’s end state as a function of r∗ and σ/M . A boundary
79
between parameter values yielding a quasi-steady or runaway growth end state clearly
exists. To understand this coupling, we utilize a probabilistic approach to predict the
evolution of a dune field.
5.4.1. Threshold between end states
As it is not clear how to estimate the probability that a dune field will enter a
specific end state, we instead ask how probable it is that a single collision involving
an individual downwind dune of mean influx size (M) will involve an upwind influx
dune of size a such that r = a/M > r∗, or a > Mr∗? As discussed in subsection
5.2.3, interactions involving r > r∗ push the dune field towards a quasi-steady end
state because the collision results in more similarly-sized dunes.
The reason that the probabilistic outcome of an individual dune collision should
be related to the predictions of a dune field’s end state is results from the nature of the
runaway growth end state: the runaway growth end state involves a small number of
dunes becoming substantially larger than the rest of the dune field through collisions.
If the probability is low that interactions between subsequent influx dunes (i.e., near
the beginning of the dune field) will yield dunes more similar in size (r > r∗), then it
is more likely that a single dune can eventually become disparately large enough to
enter into runaway growth. Furthermore, as runaway growth involves a dune growing
through collisions, the initial size that we consider is somewhat arbitrary – so we
consider the probability with regards to the mean sized influx dune.
We hypothesize that calculating the probability that a > r∗M for the lowest
r∗ which yields runaway growth will yield insight about the boundary between the
possible simulation end states. The results of many simulations are shown in Table
5.1, where it can be seen that the probability at the boundary between the two end
states is consistently very large. We can see in Figure 5.12 that as long as the size
distribution of the influx dunes is not overly wide (σ < 0.5M), even with a 90%
80
probability that any single interaction will involve a mass ratio r > r∗ and will yield
more similarly-sized dunes, the simulation will still achieve runaway growth.
dune size mean threshold probability atrange [m2] size (M) r∗ threshold r∗10− 70 40 0.4 90%25− 55 40 0.6 100%35− 45 40 0.9 90%25− 175 100 0.4 90%50− 150 100 0.5 100%80− 120 100 0.8 100%
Table 5.1. Table showing the probability that a collision involving a mean-sizedinflux dune will yield more similarly-sized dunes, for simulations with different influxranges, and at the lowest r∗ yielding runaway growth. If the dune range given is theuniform distribution [c, d], then the mean size M = (c + d)/2, and the probabilityP (a > Mr∗) = 1−P (a < Mr∗) = 1− (Mr∗− c)/(d− c) = (M −Mr∗−
√3σ)/σ
√12.
Physical implications of this probabilistic analysis are discussed in subsection
5.4.3.
5.4.2. Influence of other model parameters
We have shown that whether or not a dune field model achieves a patterned structure
depends intimately on the interaction dynamics and dune formation. Thus, both of
these processes need to be well constrained before a model can be compared to a
physical dune field. If one is only interested in predicting the end state of the sim-
ulation, however, just one parameter in dynamics (the interaction function crossover
value) and and one parameter in dune initialization (the influx condition’s standard
deviation/mean ratio) need to be constrained. Theoretically, if an actual dune field is
patterned and appears to be stable, then finding one of these two parameters should
also yield information about the other.
Many other structural factors do affect the timescale over which the simulation
achieves its end state, however, which by extension affects dune size and interdune
81
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
crossover value
influ
x du
ne p
opul
atio
n σ
/ M
ratio
Runaway growth
Quasi−steady state
90%
100%
Figure 5.12. Plot showing the model end state, as a function of the interactionfunction crossover value and the standard deviation/mean ratio (uniform distribu-tion). Notice the clear boundary between the regions corresponding to each endstate: quasi-steady state (squares) vs. runaway growth (circles). This boundaryis consistent between simulations with a mean influx dune size of 40m2 (small pointmarkers) and 100m2 (large point markers). Additionally, it is not linear, but is roughlybounded (when standard deviation < 0.5 mean) by the trendlines corresponding toa 100% (dashed) and 90% (solid) probability that a collision involving a downwindmean-sized dune will have a mass ratio r > r∗ and thus result in two similarly sizeddunes.
82
spacing (Table 5.2). This means that unless a physical dune field has achieved its end
state, these other factors (such as dune velocity and initial conditions) must also be
constrained to reliably compare simulation results with observations.
Structural element EffectInteraction function: Form (e.g., steepness) timescale
Extent of coalescence zone timescaleMultiple regions of f(r) > r timescaleCrossover value end state
Initial conditions: negligible as long as notcontaining dunes overlylarge compared to the in-flux, and sufficient time isgiven for them to be as-similated/run ahead
Influx condition: Mean value timescaleStandard deviation/mean end state
Table 5.2. A summary of different dune field model components, for a model withsemi-infinite boundaries, and the effect each component has on the model’s end stateresults. Any component which affects timescale will also affect dune size and spacingevolution. In this study, we were primarily interested in the simulation’s end state.
5.4.3. Physical implications
As was shown in subsection 5.4.1, a dune field will achieve runaway growth even if a
high probability (90%) exists that an average collision near the spatial beginning of
the dune field will result in similarly sized dunes (r > r∗). An even larger probability
is needed for the field to achieve a quasi-steady end state – which relates to a very
careful coupling between the influx condition and the interaction function crossover
value; given this, it appears surprising that most observed dune fields do not appear
to be in a state of runaway growth.
Assuming the premise that the dune field model presented in Section 5.3 ade-
quately captures the results of dune collisions, there are three extreme explanations:
83
1. Collisions are the dominant stabilizing mechanism for dune fields. Thus, for
most dune fields to appear patterned, the physical system – both the environ-
mental conditions which influence dune formation and the interaction dynamics
– must naturally fall into the small window needed for this to occur.
2. Collisions may play a role in redistributing sand from large dunes to small
dunes, but the model is incomplete. Other processes (e.g., interdune sand flux
or intrafield dune nucleation) or factors (e.g., variations in influx) are needed
to increase the stability of patterned dune fields.
3. Patterned dune fields are not stable landforms. The timescale over which a
dune field destabilizes, however, is very long compared to the timescales over
which its environment changes.
The first point is a plausible explanation for barchan fields, as numerical simula-
tions of barchan dune interactions yield interaction functions with very low crossover
values, which means the influx distribution does not need to be tightly constrained
for the dune field to possibly become patterned. For example, Duran et al. (2005)
found an r∗ ∼ 0.12 (volume ratio) for zero-offset collisions between barchans (Fig.
5.6). Using a crossover value that low in this model (translated to r∗ = 0.16 for the
cross-sectional area ratio by Lee et al. (2005)), and assuming a uniformly distributed
area influx starting with 10m2 (∼ 1m height), the upperlimit on the influx distri-
bution can be as high as 110m2 (∼ 5m height) and the system will still achieve a
quasi-steady state. Additionally, even if the crossover value is higher, it seems rea-
sonable to assume that dune initialization should result in a narrow range of dune
sizes, so still yield a quasi-steady end state.
In Section 5.2, we showed, however, that simulations of transverse dune collisions
yield r∗ = 1. In this case, if transverse dune fields do achieve stable patterned
structures, the second explanation must be correct. Alternatively, the third point is
correct (or the original premise is wrong).
84
The second explanation (that the model is incomplete) needs to be carefully eval-
uated, as size- or age-dependent dune field stabilizing processes do occur. For ex-
ample, large barchan dunes can be destabilized through wind variations and dune
collisions, as long-wavelength perturbations form on the flanks of these dunes and
break away. This prevents any dunes from becoming overly large and increases the
likelihood that a dune field will remain patterned. Intrafield dune destabilization has
been studied in the field (Elbelrhiti et al. 2005, Gay 1999), in the laboratory (Endo,
Taniguchi & Katsuki 2004), and in three-dimensional continuum simulations (Duran
et al. 2005, Katsuki, Nishimori, Endo & Taniguchi 2005, Elbelrhiti et al. 2008).
The results of dune collisions may also depend on local conditions, such as the age
of dunes involved or the type of sand making up the dunes. In Besler (2002), on-going
dune collisions were studied in different fields in the Libyan desert, and the apparent
collision results in each locale were compared with the dunes’ granulometrics. In that
study, it was hypothesized that a downwind dune made of softer and finer grains was
more likely to coalesce with the colliding upwind dune, while a downwind dune made
of more compacted and coarser grains was more likely to have a collision result in
ejection. As younger/smaller dunes are more likely to contain softer, finer grains,
and older/larger dunes are more likely to contain more compacted, coarser grains
(Besler 2005), this would mean that large dunes would be less likely to coalesce and
enter runaway growth.
The rate of sand redistribution through these processes and/or their effect on the
dune field’s interaction function, however, still needs to be quantified through numer-
ical simulation, observation, and/or laboratory experiments, before these processes
can be included in a dune field model.
The final explanation is in reference to the fact that, in this model, the timescale
of collisions was ignored and the rate of migration and dune injection were arbitrarily
specified. This is acceptable as long as we are concerned only with the dune field’s
end state. However, if the timescale over which a dune field achieves runaway growth
85
is overly long, environmental conditions may vary or the dune field may run into a
physical boundary long before the dune field will reach this state. Currently, there is
no reason to expect that the timescale over which runaway growth occurs should be
very long. In this study, our choice of parameters yielded timescales on the order of
a century (which is consistent with estimates by Hersen et al. (2004)).
However, long timescales may play a role if the actual distribution of dune sizes
is strongly-peaked. With simulations that were run with Gaussian influx dune size
distributions (with comparable σ/M as were used with uniform distributions, and
with a low-end cutoff at 1m2), runaway growth occurred at lower crossover values
(Fig. 5.13), but after much longer time periods (10 − 100 times longer). This was
unexpected, because we had hypothesized that a peaked distribution could have the
same standard deviation/mean ratio value as a uniform distribution, and be far less
likely to have a sufficient number of small r interactions for the dune field to achieve
runaway. The ‘tails’ of the distribution ended up being more influential than the peak,
however – very large or very small dunes had a very low probability of being injected
into the dune field, but after a very long time it was more likely that an apparently
‘stable’ patterned dune field would become destabilized and enter runaway growth.
The peaked nature of the distribution did exert a ‘stabilizing’ influence on transi-
tory dune field dynamics – in a few cases with intermediary r∗ values, a field would
appear to switch from quasi-steady state to runaway growth and then back. Addition-
ally, when simulations were run with tail-less Gaussian distributions (which is a more
physically realistic distribution), the observed decrease in the threshold crossover val-
ues disappeared. The impact of using a realistic influx dune size distribution should
be more thoroughly studied in the future.
86
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.05
0.1
0.15
0.2
0.25
0.3
0.35
crossover value
influ
x du
ne p
opul
atio
nσ
/ M
ratio
Quasi−steady state
Runaway growth
Figure 5.13. Plot showing the model end state, as a function of the interactionfunction crossover value and the standard deviation/mean ratio (Gaussian distribu-tion). Notice that the boundary between the regions corresponding to each end state(roughly demarked by the solid trendline): quasi-steady state (squares) vs. runawaygrowth (circles), is at lower crossover values as compared to the results with uniformdistribution (dashed trendline). However, simulations required 10−100x longer timesto achieve an end state. Additionally, the left shift in the boundary disappears if thetails are removed from the Gaussian distribution. As in Figure 5.12, the size of thepoint-markers denotes the mean of the dune size distribution.
87
5.4.4. Possible future model improvements
As discussed in the second point in subsection 5.4.3, some additional physical pro-
cesses may need to be included before the model can provide a sufficiently complete
picture of dune field evolution. For example, the model may be improved by including
(i) sand flux between and during collisions, (ii) dune destabilization between and dur-
ing collisions, and (iii) variations in the influx condition or interaction function. Ad-
ditionally, if the model is to be expanded to consider evolution of a three-dimensional
dune field, dunes would be able to split into more than two dunes after a collision
(Duran et al. 2005, Katsuki, Nishimori, Endo & Taniguchi 2005).
As long as the processes and effects considered are assumed to occur during a
collision (such as sand flux, dune destabilization, and/or a higher number of resultant
dunes), then the model would not need to significantly change in structure. The effects
of sand flux and dune destabilization during a collision could simply be added into
a higher-dimension interaction function (e.g., the total size of the dunes and the size
ratio after collision could be a function of the size ratio and total sizes of the dunes
before collision), and the interaction function could reflect the formation of more
than two dunes by calculating multiple after-collision size ratios. Thus, although the
function would doubtlessly be much more complex, it is unlikely that the nature of its
effect on the dune field’s end state would significantly change from what is presented
in this study; i.e., it is likely that the “crossover surface” would still be the only
influential component of the interaction function in the analysis about the end state
of a dune field.
Another possible addition to the model would be to include temporal or spa-
tial variations in the interaction function (e.g., because of granulometric sand type
changes in the dunes (Besler 2002)), or temporal variations in the influx condition
(e.g., because of changes in local vegetation or sand supply). In this case, the range
of possible r∗ and/or σ/M values, and the rates of parameter change would need to
88
be incorporated into the analysis, as those factors could push a dune field between
end states. For example, if the interaction function crossover value could widely vary
over short time/spatial scales, a dune field that would be expected to enter runaway
growth at the mean crossover value could in fact be stabilized as large dunes would
break up before becoming sufficiently larger than the surrounding dunes. If, instead,
the influx dune population’s standard deviation changed over long time periods, an
apparently patterned dune field could be destabilized.
Furthermore, if the variation in range and period of these parameters were properly
coupled, it should be possible to see oscillation between the two end states. In fact,
all of these different types of behavior (apparently runaway → patterned, apparently
patterned → runaway, and oscillatory) have been observed in simulations with a
Gaussian distribution of influx sand dunes (see subsection 5.4.3).
To complete the model, it may also be necessary to allow dune evolution between
collisions – such as through sand flux or dune destabilization. These types of inter-
collision processes, however, would be more problematic to add into the model, as
now dune evolution is partially uncoupled from the collisions. New analysis would
needed to carefully determine inter- and intra-collision dune size evolution functions,
and related parameter(s) would need to be constrained.
Finally, if we are to include temporally or spatially varying parameters, and/or
inter-collision dune evolution in the model, then timescale becomes an important
concern. Dune evolution can now occur continuously (e.g., through intra- and inter-
collision sand flux), semi-periodically (e.g., with a varying influx condition), or even
stochastically (e.g., storm-caused destabilization of large dunes). With these different
timescales, superimposed periodicities between dune growth and destabilization may
occur, possibly resulting in pseudo-periodic or chaotic switching between the runaway
growth and quasi-steady states.
As we can see, a more “complete” model can quickly become much more complex
– and interesting. However, to properly include these processes in the model, identify
89
influential parameters, and be able to derive predictions about a particular dune field,
a far better understanding of these processes is needed. Thus, experimental and field
work is vital in providing a quantification of the timescales and effects these different
processes have on dune and dune field evolution.
90
6. INFLUENCE OF BEDROCK TOPOGRAPHY
As we have shown, interactions between airflow, sediment transport, and topography
have a large impact on the formation and distribution of aeolian landforms, such
as dunes; Chapters 2 and 3 described in detail how dunes are formed through a
feedback between dune topography and wind-induced sediment transport. However,
it is not well-understood how non-flat (non-erodible) bedrock topography will affect
dune formation and evolution; despite (or perhaps because of) this, geological studies
will often attribute strange dune forms to the effect of terrain. For example, the shape
and orientation of Antarctic ‘whaleback’ dunes was attributed to underlying moraine
structure (Calkin & Rutford 1974, Selby et al. 1974) (a study by Bristow et al. (2009)
showed that this was not the case).
In this chapter, we investigate the effect of non-flat bedrock topography on isolated
dune size and migration. Those results will then be extended to the effect that terrain
will have on binary dune collisions, and related dune field evolution.
6.1. Physical Description
Numerical dune simulations are generally run with a flat bedrock layer, whose only
effects are to halt erosion beyond a baseline (e.g., h = 0) and to enhance saltation over
exposed bedrock. However, dune fields are found in non-flat environments: martian
dune fields are often found inside of craters (Fenton et al. 2005) and mountains and
valleys shape the dune fields on Earth (Gaylord & Dawson 1987, Wiggs et al. 2002)
and Mars (Bourke, Bullard & Barnouin-Jha 2004). However, it is not well-understood
how to include relevant topography within the model without greatly complicating
the simulation or obscuring features of interest.
Additionally, detailed topographic information is often not available for inclusion
91
in the models. In these cases, an understanding of the connection between dune mor-
phology and underlying topography could be used to reverse-engineer estimates of
terrain slopes based on observed dune morphology. For example, qualitative analysis
of this type is currently being done for Titan. We have almost no topography informa-
tion about this moon of Saturn, but comparative analysis between dune terminations
on Titan and dunes in the Namibian desert has yielded information about the sign
of topographical slopes (i.e., topographic highs vs. lows). In some places, this dif-
fered from topographical inputs to global atmosphere circulation models (Radebaugh
et al. 2010); analysis of dune forms also yielded opposite wind directions (Radebaugh
et al. 2008) from those predicted by the atmosphere model, but changing the topo-
graphical inputs yielded more consistent results (Radebaugh et al. 2010).
Here we present preliminary attempts to study the influence topography exerts
on the migration and shape of an isolated dune. We will extend these results to
qualitatively estimate effects within large-scale and more complicated dune fields,
but note that current numerical experiments have the following limitations:
• As the simulation is two-dimensional, the perturbations considered are perpen-
dicular to wind flow, causing obstruction of sediment transport or sheltering
effects. We do not consider the effect of channeled airflow or other three-
dimensional effects.
• To more closely identify and quantify the influence of topography, we restrict
ourselves to only small and simple perturbations: a Gaussian hill or an upwards
or downwards step.
Future studies should consider more complicated underlying topography, and will
require validation through quantatitive comparison with observed dune fields.
92
6.2. Influence of Terrain on Isolated Dune Evolution
Numerical experiments in non-dimensional space were used to investigate how dune
shape and migration dynamics changed as a dune moved over topography – a Gaus-
sian hill of specified height and standard deviation or an upward/downward step of
specified height (Figure 6.1). Simulations were initialized with isolated dune profiles
that were found to be steady-state over a flat surface, with dune area of 100-400.
Results from these control simulations with dunes of size 120, 280, 400 (the results
generally shown in figures) are listed in Table 6.1.
−100 0 100 200 300 400 5000
0.5
1
1.5
−100 0 100 200 300 400 5000
0.5
1
1.5
Figure 6.1. Diagrams showing simulation ground initialization for investigatingthe influence topography will have on dune migration and evolution. We focus on abedrock Gaussian hill (top): y = A exp(-(x − 100)2/2σ2) or upward/downward step
(bottom): y =
{A x ∈ [100, 300]
0 otherwise.. The Gaussian hill example has height A = 1
and standard deviation σ = 10; the step example has height A = 1.
6.2.1. Over a smooth hill
As shown in Figures 6.2-6.4 (0.5, 1 and 5-high Gaussian hills, respectively), the dune’s
shape and migration rate varied both before and after encountering the hill, within
93
Dune area height length migration velocity120 7.53 43.48 0.3401280 11.74 63.10 0.2227400 14.09 74.33 0.1864
Table 6.1. Table showing dune shape and migration values over flat topography.
a zone of perturbation. After encountering the hill, the dune adjusted back to its
original shape and migration rate.
6.2.2. Over upward and downward steps
As was found with the Gaussian hill, the dune’s shape and migration rate were
changed upon encountering a step, and then readjusted within a zone of pertur-
bation. Figures 6.5-6.7 show changes in dune migration rate, height, and length for
steps of 0.5, 1 and 2-high, respectively.
Upon encountering the upward step, the lower portion of the dune became caught.
This caused the dune crest to migrate faster, as effectively a smaller dune (the upper-
portion only) was migrating forward, over the step. This caused the dune height
to decrease and the dune length to increase. However, as the upper-portion moved
forward, the lower portion of the dune was then exposed to the wind and began to
also move over the step, causing the dune length to decrease (to less than the regular
amount). Finally, after some distance from the step (which increased with both dune
and step size), the dune adjusted to nearly its original shape and migration rate.
The reverse happened upon encountering the downward step. The dune initially
slowed down when it reached the step, as sand was sheltered behind the step. As the
sand pile behind the step grew to be taller than the step, the upper portions were
exposed to the wind and began to migrate with a dune migration velocity higher
than the original velocity. Eventually, most of the sand reaccumulated within the
dune and it adjusted back to a steady shape and migration rate. When the step was
94
0 100 200 300 4000.33
0.34
0.35
0.36Migration velocity
A =
120
0 100 200 300 4007
7.5
8
8.5Dune height
0 100 200 300 40040
42
44
46Dune length
0 100 200 300 400
0.98
1
1.02
norm
aliz
ed
0 100 200 300 4000.960.98
11.021.041.06
dune crest location0 100 200 300 400
0.920.940.960.98
11.021.04
0 100 200 300 4000.215
0.22
0.225
0.23
A =
280
0 100 200 300 40011
11.5
12
12.5
0 100 200 300 40058
60
62
64
66
0 100 200 300 4000.18
0.185
0.19
0.195
0.2
A =
400
0 100 200 300 40013.5
14
14.5
15
0 100 200 300 40065
70
75
80
Figure 6.2. Plots showing the changes induced in dune migration rate, height, andlength by a 0.5-high Gaussian hill located at 100, with standard deviations of 10and 20 (thick lines). Dunes plotted had sizes of 120, 280, and 400; the bottom plotscontain values normalized by the values found with no topography (Table 6.1).
95
0 100 200 300 4000.32
0.33
0.34
0.35
0.36Migration velocity
A =
120
0 100 200 300 400
7
7.5
8
8.5
Dune height
0 100 200 300 40038
42
46
Dune length
0 100 200 300 4000.95
1
1.05
norm
aliz
ed
0 100 200 300 400
0.95
1
1.05
1.1
dune crest location0 100 200 300 400
0.9
0.95
1
1.05
0 100 200 300 4000.21
0.22
0.23
0.24
A =
280
0 100 200 300 40011
11.5
12
12.5
13
0 100 200 300 40055
60
65
70
0 100 200 300 4000.175
0.18
0.185
0.19
0.195
A =
400
0 100 200 300 40013
14
15
16
0 100 200 300 40065
70
75
80
85
Figure 6.3. Plots showing the changes induced in dune migration rate, height, andlength by a 1-high Gaussian hill located at 100, with standard deviations of 10 and20 (thick lines). Details are as described in Figure 6.2.
96
0 100 200 300 4000.2
0.4
0.6
0.8
1Migration velocity
A =
120
0 100 200 300 4000
5
10
15Dune height
0 100 200 300 4000
50
100
150Dune length
0 100 200 300 4000.5
1
1.5
2
norm
aliz
ed
0 100 200 300 400
0.5
1
1.5
dune crest location0 100 200 300 400
1
1.5
2
0 100 200 300 400
0.2
0.25
0.3
0.35
A =
280
0 100 200 300 4005
10
15
20
0 100 200 300 4000
50
100
150
0 100 200 300 4000.1
0.2
0.3
0.4
0.5
A =
400
0 100 200 300 4005
10
15
20
0 100 200 300 40050
100
150
Figure 6.4. Plots showing the changes induced in dune migration rate, height, andlength by a 5-high Gaussian hill located at 100, with standard deviations of 10 and20 (thick lines). Strange behaviors occur when the dune height is comparable to thetopography height; for example, the 120 dune (7.5-high) appears to move backwardsas it encounters the 5-high hill, temporarily morphing into a two humped feature.Neither the 120 nor 280 dunes survive migration over the steep 5-high, 10-standarddeviation hill (they both became stuck behind the hill, and slowly leaked sand), andeven the 400 dune loses sand (height drops to ∼ 0.75 original value). Plot legend isas described in Figure 6.2.
97
much smaller than the dune (e.g., Figure 6.5), the dune nearly-recovered its original
shape and size. When the step was comparable to the dune height, a large fraction
of the dune mass became trapped behind the backward step, causing the final dune
size to be much smaller; this is especially visible in the differences between dunes in
Figures 6.6-6.7.
6.2.3. Effect on isolated dune evolution
With both the Gaussian hill and steps, the change in dune migration speed (and asso-
ciated changes in dune shape) depended primarily on the topography to dune height
ratio (Figure 6.8); we found a near-linear relation between the maximal fractional
change in velocity and this height ratio. Generally, the maximum change in velocity
over an upward step was an increase, and the maximum change over a downward
step and a Gaussian hill was a decrease. Exceptions occurred when the perturbing
topography’s height was large; in these cases, sand was lost to the sheltered lee of the
topography or due to leakage during the encounter, which resulted in a smaller dune
size and higher migration velocity.
We also investigated the length of the zone of perturbation (both before and after
the topography; Figure 6.9) to investigate the spatial extent of transitory changes in
dune shape and migration rate. We found that:
• The distance over which the dune was perturbed in shape and size before a
step was independent of dune size and increased with the perturbation height.
Before the Gaussian hill, the dune was perturbed much earlier (roughly one dune
length); perturbation lengths in this case exhibited nonlinear dependencies on
dune size and perturbation shape.
• After encountering either a step (upward or downward) or a Gaussian hill, the
distance over which the dune adjusted back into a steady-state shape seems to
98
0 200 4000.32
0.34
0.36Migration velocity
A =
120
0 200 4006.5
7
7.5
8
8.5Dune height
0 200 40040
45
50
55Dune length
0 200 4000.96
0.98
1
1.02
norm
aliz
ed
0 200 400
0.95
1
1.05
dune crest location0 200 400
1
1.1
1.2
0 200 4000.215
0.22
0.225
0.23
A =
280
0 200 40011
11.5
12
12.5
0 200 40050
60
70
80
0 200 4000.18
0.19
0.2
A =
400
0 200 40013
14
15
0 200 40070
80
90
Figure 6.5. Plots showing the changes induced in dune shape and migration rateby 0.5-high steps, upward at 100 and downward at 300. Dunes plotted had sizes of120, 280, and 400; the bottom plots contain values normalized by the values foundwith no topography (Table 6.1).
99
0 200 4000.3
0.35
0.4Migration velocity
A =
120
0 200 4006
7
8
9Dune height
0 200 40030
40
50
60
70Dune length
0 200 400
0.95
1
1.05
norm
aliz
ed
0 200 400
0.9
1
1.1
dune crest location0 200 400
1
1.2
1.4
0 200 4000.2
0.22
0.24
0.26
A =
280
0 200 40010
11
12
13
0 200 40050
60
70
80
90
0 200 4000.17
0.18
0.19
0.2
A =
400
0 200 40012
14
16
0 200 40060
80
100
Figure 6.6. Plots showing the changes induced in dune shape and migration rateby 1-high steps, upward at 100 and downward at 300. Plot legend is as described inFigure 6.5.
100
0 200 4000.25
0.3
0.35
0.4
Migration velocity
A =
120
0 200 4004
6
8
10Dune height
0 200 40020
40
60
80Dune length
0 200 400
0.9
1
1.1
norm
aliz
ed
0 200 400
0.8
1
1.2
dune crest location0 200 400
0.8
1
1.2
1.4
1.6
0 200 400
0.2
0.25
0.3
A =
280
0 200 4008
10
12
14
0 200 40040
60
80
100
0 200 4000.16
0.18
0.2
0.22
A =
400
0 200 40010
12
14
16
18
0 200 40060
80
100
120
Figure 6.7. Plots showing the changes induced in dune shape and migration rateby 2-high steps, upward at 100 and downward at 300. Plot legend is as described inFigure 6.5.
101
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.05
0.1
0.15
0.2
0.25
perturbation height/dune height
max
imum
frac
tiona
l cha
nge
in v
eloc
ity
Figure 6.8. The maximal change in dune velocity (either increase or decrease) dueto topography, either a step (black; * = upward, o = downward) or a Gaussian hill(gray; � = standard deviation of 10, � = 20). The line markers denote perturbationheight: 0.5 (solid), 1 (dashed), 2 (dash-dotted), and 5 (dotted); they connect thevalues for different dune sizes: 120-280-400 (left to right within a sequence).
depend on both the perturbation size and shape. In most cases, this distance
scales roughly with dune length: ∼ 2−3 times the pre-perturbation dune length.
6.3. Implications for Field Evolution
Based on these preliminary numerical experiments of how terrain changed the evolu-
tion and behavior of an isolated dune, we begin to estimate the influence of topography
on dune field evolution.
6.3.1. Effect on dune collisions
Dune migration over topography will alter dune collision dynamics and the interac-
tion function due to induced changes in dune migration velocity. Changes in dune
102
0 0.1 0.2 0.30
0.5
1
1.5
2
2.5
3
3.5
4
4.5step
perturbation height/dune height
pertu
rbat
ion
zone
leng
th/d
une
leng
th
0.1 0.2 0.3 0.4 0.5 0.6 0.7
gaussian hill
Figure 6.9. The perturbation zone length, before (black) and after (gray) thetopography normalized by the dune length (before perturbation), as a function ofperturbation height normalized by the dune height. The left plot shows the effectsof an upward step (*) and downward step (�), while the right plot shows the effectsof a Gaussian hill with standard deviation of 10 (o) or 20 (�). The line markersdenote perturbation height: 0.5 (solid), 1 (dashed), 2 (dash-dotted), and 5 (dotted);they connect the values for different dune sizes: 120-280-400 (left to right within asequence).
velocity have a large influence on the amount of sand exchanged throughout a dune
collision as this depends primarily on the timescale of the collision (as described in
subsection 5.2.2). In this study, we consider only cases where topography is located
downwind of colliding dunes; topography located upwind of dunes will perturb the
shear stress exerted on the dunes (see subsection 5.1.2), but will not otherwise affect
dune collisions.
Generally we found that the presence of topography will increase the likelihood
of a dune collision to result in coalescence. In simulations run with Gaussian hills
and downward steps, the non-flat topography caused a decrease in dune migration
rate, causing the ejected dune to be slowed. This allowed the dunes to interact a
second time, and for sand to be moved back from the smaller dune to the larger.
103
Figure 6.10 illustrates a clear example of this: this collision resulted in ejection over
flat terrain, but a downwind encounter with a Gaussian hill caused further sand
exchange, resulting in coalescence and net sand loss. In other simulations where the
topography was large relative to the ejected dune, sand was lost from this dune even
if it did not re-interact with the upwind, larger dune; in either case, the downwind
dune became smaller, making later coalescence more likely.
0
100
200
300
400
0
100
200
300
400
dune
are
a
time (increasing to right)
upwind dunedownwind dune
secondinteraction
A
B
collision
collision
EJECTION
EJECTION
COALESCENCE (with some sand loss)
Figure 6.10. These plots show the size of colliding dunes (120 with 240) throughouta collision. In (A) the dunes collide on flat terrain: they briefly merge and then asmall dune is ejected. (B) shows the perturbative effect of topography, as the presenceof a 1-high, 10-standard deviation Gaussian hill downwind of the collision slows theejected dune sufficiently to cause a second interaction, resulting in coalescence andnet sand loss.
If it is possible for sand to be moved from the larger dune to the smaller, this
would instead result in a pair of dunes that are more similar in size. This could
perhaps occur if the upwind dune become caught on large topography (such as a hill
or upward step) and leaked sand to the smaller downwind dune. However, within
these two-dimensional simulations, the topography would have first interacted with
104
the smaller downwind dune, and would have exerted a greater effect due to the larger
relative size. Thus, the net change in the downwind dune’s size should still be to
decrease in size.
In three dimensions, it may be possible for the smaller dune to bypass topography
which catches the upwind larger dune. Additionally, topography that is less wide than
the dune (and thus will exert three-dimensional effects) can perhaps cause a larger
dune to break into several pieces. This has been shown to occur within barchan dunes
due to wind variations (Elbelrhiti et al. 2005).
6.3.2. Effect on dune field pattern formation
Binary collisions within two-dimensional fields and occurring around and over non-
flat topography appear to result in more cases of coalescence, or at least result in
dunes of more disparate sizes. Individual dune collisions that result in sand transfer
from the smaller to the larger dune pushes the system toward run-away growth (as
discussed in subsection 5.4.1). Thus, it appears that topography, in affecting dune
collisions, will have a destabilizing influence on pattern formation within transverse
dune fields.
However, this influence can perhaps be dampened. For example, as dunes recover
from their interaction with topography over a few dune-lengths (Figure 6.9), we would
expect topography to exert a destabilizing influence only within closely-packed dune
fields (where dune spacing is comparable to dune length). In dune fields with large-
interdunal spacings, the influence topography will exert on dune collisions may be
minimized. Alternatively, the development of sediment-rich interdune areas (such as
those found in most transverse dune fields) can insulate the dunes from large bedrock
topography, dampening the influence of terrain even within closely-packed fields.
Thus, bedrock topography may only affect dune collision dynamics under specific
conditions. If these conditions occur only during an early period of transverse dune
105
field evolution, this could cause an initial enhancement of coalescence within the field;
the influence of topography may decrease as the field matures and allow the field to
stabilize (under the influence of other processes).
6.3.3. Inclusion of topography in the dune field model
As the primary effect of terrain on dune collisions and dune field evolution is related
to variations in dune migration velocities, it should be possible to include topography
within the dune field model outlined in Section 5.3 by:
• Letting the dune migration velocity (between collisions) be a function of prox-
imity to terrain and the ratio of dune height to topography height. Simplifying
assumptions will be needed to accomodate dependecies on topography shape.
• Modifying the interaction function to reflect changes in relative dune velocity
when terrain is also involved in the collision. As in the first point, this will
depend on the relative size of the topography with respect to the interacting
dune and possibly on the shape of the topography.
• Sand leakage will need to be considered between and during dune collisions,
when a small dune encounters large topography. Lost sand can generally be
assumed to accumulate on the downwind dune (in two-dimensions).
Many more simulations and studies are needed to determine simple relations that
will reflect the way that dune migration speed changes based on perturbation height
(relative to dune height) and shape, and the distance over which terrain will exert an
influence is felt. These preliminary results, however, do indicate that simple relations
may be found for how the dune migration velocity varies with topography height (rel-
ative to dune height). The most complicated quantity appears to be the perturbation
zone length.
106
These results also indicate that simple two-dimensional studies will only show
enhanced coalescence within dune fields. More complicated dune field evolution pro-
cesses (such as sand accumulation in the interdune areas) and three-dimensional ef-
fects may need to be considered to determine if topography can exert a stabilizing
influence on dune field evolution.
107
7. AN APPLICATION: MARTIAN POLAR DUNES
In this chapter, we investigate the influence of reversing wind directions, diffusive
processes, and ice cementation on dune slope evolution. The aim is to quantitatively
connect specific morphologies to processes and environmental parameters, which will
aid in the interpretation of actual dune forms.
Figure 7.1. An image (HiRISE PSP 010269 2620; 81.7N,133.6E)of an example ofthe break in lee slope seen in many martian polar dunes. The appearance of a ∼10mwide “bright ribbon” along the dune brink is due to a lack of ripples and increasedslope on the upper portion of the lee slope, relative to the lower portion.
This work was motivated by our observation that martian dunes located in the
mid-latitudes and polar regions exhibit a break in slipface slope (Figure 7.1), which is
not found elsewhere (Figure 7.2, 7.3). Within terrestrial dunes, this type of slipface
108
morphology forms due to wind reversals (Koster & Dijkmans 1988, Burkinshaw et al.
1993, Bishop 2001, Bristow et al. 2010). Due to the apparent latitudinal control on
where this feature is observed, we hypothesize that dune geomorphology may also be
affected by cold climate processes, such the formation of an ice cemented core and/or
increased downslope sand transport.
��
���� ����
��
Figure 7.2. HiRISE images of other lee slopes seen on martian dunes.Image (a) shows examples of sharp, clean brinks and smooth lee slopes(PSP 010413 1920; 20N,79E). Image (b) shows examples of shallow and round leeslopes (PSP 006716 1220; 60S, 340E).
7.1. Physical Description
Antarctic dunes are studied as possible terrestrial analogues of martian polar dunes,
due to similarities between their cold and arid formative environments. They also
appear to share many morphological details, such as denivation features: fans, slumps,
and visible ice layers have been observed within both cold-climate terrestrial (Bourke
et al. 2009, Calkin & Rutford 1974, Steidtmann 1973) and martian dunes (Bourke
2005, Horgan et al. 2010). Within Antartic dunes, these features form due to the
109
Figure 7.3. HiRISE images of dune fields were inspected to located dunes with“clean” and sharp slipfaces and crestlines (red), highly-eroded slipfaces and roundedcrestlines (blue), and slipfaces with an apparent break in slope (green). Black dotsdenote dunes where viewing conditions (illumination and/or frost cover) made itdifficult to discern the condition of the slipface.
inclusion of inter-bedded sand, snow, and/or ice deposits (Bourke et al. 2009, Calkin
& Rutford 1974). There is much evidence for subsurface water ice on Mars, in both
the mid-latitudes and polar regions. In the North Polar Sand Sea on Mars, neutron
current measurements and modeling results imply the existence of an ice-rich (and
immobile) underlying topography with a relatively desiccated upper-layer (∼ 6cm
depth; Feldman et al. 2008), and recently-formed impact craters in the mid-latitudes
have exposed near-surface pure ice (Byrne et al. 2009).
Understanding how cold climate processes will change dune evolution has impor-
tant implications for present-day dune evolution on Mars, as such processes will slow
landform migration. In cold-climate environments, dunes often are stabilized due
to coverings of snow, ice-cementation, and possible increases to the threshold shear
velocity of sand motion due to increased humidity (Lindsay 1973, Selby et al. 1974).
Reversing (seasonal) winds will also slow dune migration, as the dune must now ad-
just its shape before migrating (Figure 7.4; Burkinshaw et al. 1993). Comparison of
110
aerial photographs of Antarctic reversing dunes yielded a 40yr (1961-2001) average
migration rate of 1.5m/yr (Bourke et al. 2009), while dating of stratigraphy within
the same dunes yielded average migration rates of 0.05-1.3m/yr over 1300yrs (Bristow
et al. 2010). This is much slower than the tens of meters per year migration rates of
similarly-sized desert dunes (Cook et al. 1993, Gay 1999).
Figure 7.4. Dune profiles measured over a 7m high African transverse dune duringthe winter season (March to September), showing stages of dune reversal. During thesummer the winds blow from the east, and during the winter from the north-west.Image is from (Burkinshaw et al. 1993).
7.2. Influence of Reversing Winds
Winds that change direction by 180o will yield reversing dunes. These dunes can be
differentiated from transverse or barchan dunes (which form in unidirection winds,
as described in subsection 2.1.1) due to their steeper and/or non-smooth slopes that
form as the stoss slope becomes the lee slope, and visa versa. On the new lee slope,
free-falling sand piles up at the angle of repose on the upper-portions of the previously-
stoss slope; on the new stoss slope, an apron of lower slope forms at the base of the
previously-lee slope. Reversing dunes can be found within terrestrial desert (Bishop
2001, Burkinshaw et al. 1993) and cold climate dune fields (Koster & Dijkmans 1988,
Bristow et al. 2010).
111
A study by Fenton et al. (2005) showed the probable existence of reversing dunes
in Proctor crater (47S, 30E) on Mars. These dunes contain slipfaces with three orien-
tations and the kinked lee slope (Figure 7.5). Two of the slipface orientations coincide
with predicted wind directions: within this intracrater dune field, geostrophic-induced
winds blow from the west during the fall and winter, and katabatic winds blow from
the east during the spring and summer. It is likely that similar reversing winds will
be found in other intracrater dune fields and in the polar regions.
To investigate the influence that reversing winds will have on dune slope and
behavior, we numerically experimented with different period lengths (duration of a
cycle during which the wind blows to the left and then right) and ratios (the time
the wind blows to the left versus to the right). Simulation parameters for sand
flux were taken from a martian barchan dune modeling study (Table 3.1; Parteli &
Herrmann 2007). Windspeed was held constant at 1.5 times the threshold value.
Simulations were initiated with a dune formed under unidirectional winds blowing to
the right.
As observed within terrestrial dunes (e.g., Burkinshaw et al. 1993), a reversing
wind slows dune migration and steepens the dune’s slopes (Figure 7.6). The amount
of steepening depends primarily on the frequency of wind reversal (Figure 7.7, 7.8),
and only weakly on the period of wind cycling and the dune size.
7.3. Influence of Diffusive Processes
Numerical experiments showed that dune slope values also depend on assumptions
made about down-slope transport of sand (e.g., avalanching) that is modeled via
diffusion. For example, reversing wind simulations run with a diffusion coefficient
one order of magnitude higher yielded much lower slopes (Figure 7.9); this change in
slope was comparable or larger than that resulting from adjustment to a wind reversal
(Figure 7.7) or any other factors, illustrating that diffusion needs to be properly
112
Figure 7.5. MOC images showing possible reversing slipfaces on dunes at the easternedge of the Proctor crater dune field (47S, 30E). Images taken in the fall (a, MOCNA M19/00307) and spring (b, MOC NA E09/02707) show the same area with brightaccumulations on opposing slipfaces. Figures (c) and (d) illustrate the locations ofslipface brinks and bright accumulations. Figure (e) shows both sets of slipfacesand bright material. The movement of bright material is thought to be caused by aseasonal shift in wind direction. Image is taken from Fenton et al. (2005).
113
10
20Dune profiles formed in reversing winds
200 250 300 350 400 450 500
10
20
10
20
after 60 solsafter 600 solsafter 5 Mars years
wind blowing to right
3:1 ratio
9:1 ratio
Figure 7.6. Under unidirectional winds, the dune migrates 97m; under 9:1 (9/10 oftime wind blows to right, 1/10 to left) winds, 63m; and under 3:1 winds, 35m. Thedune also grows in height, as its slopes steepen. The period used was six sols.
estimated before simulation results can be compared with observed dunes. In the
physical situation, this dependence on diffusion reflects the fact that a dune’s shape
results from a competion between saltation (which piles sand and increases slope)
and diffusion (which decreases slope).
In terrestrial desert dune systems, it generally can be assumed that diffusion is
comparable in strength to saltation only within lee slope avalanches. In this case,
as discussed in subsection 3.1.5, the diffusion coefficient is chosen so that diffusion
operates on the same timescale as saltation when avalanches occur, and at a much
longer timescale (increased by 2-3 orders of magnitude) otherwise. The effective
diffusion, however, can be increased through any process that preferentially transports
material downslope, such as atmospheric turbulence, creep, freeze-thaw cycles, seismic
shaking, micro-meteorite impacts, or CO2 sublimation (Figure 7.10).
Additionally, if saltation does not occur for a large fraction of time (due to sedi-
ment or wind limits), then simulation results and model time need to be scaled via
114
0 1 2 3 4 55
10
15
simulation time (Mars yrs)
mea
n st
oss
slop
e (o )
0.02 Mars yr0.05 Mars yr0.2 Mars yr0.5 Mars yr1.7 Mars yr
0 1 2 3 4 5−30
−25
−20
−15
simulation time (Mars yrs)
mea
n le
e sl
ope
(o )
0.02 Mars yr0.05 Mars yr0.2 Mars yr0.5 Mars yr1.7 Mars yr
Figure 7.7. Simulations were run with a ratio of 3:1 for the time the wind blewto the right versus left, with different period lengths. Both stoss (upper plot) andlee (lower) slopes adjust to a mean value, with primary dependence on the ratio (notperiod length). The variance in average slope does increase with the period length,as the dune has a longer time to adjust to a wind direction before it reverses.
115
1/11/3 3/1 9/11/9
10
14
18
wind reversal ratio
angl
e (o )
left sloperight slope
lee slope limit
unidirectional to left to right
stoss slope limit
Figure 7.8. Both lee and stoss dune slopes adjust to a mean value that depends onthe frequency of wind reversals, but is nearly independent of the period length (notshown) and dune size (large markers: dunes ∼30m high; small: ∼20m.). Dashed linesshow effect of unidirectional wind, which is consistent in stoss values with observations(Burkinshaw et al. 1993, Parteli, Schwammle, Herrmann, Monteiro & Maia 2005).The mean lee slope is lower than the angle or repose as calculations include smoothingat the top and bottom of the slope. This is the cause of the apparent oversteepening ofthe lee slope – as the wind reverses direction, the smoothing-portions of the previously-lee slope (which lie at very low slopes, especially at the foot of the slipface) are swepttowards the dune and shortened.
116
1/11/3 3/1 9/11/9
6
10
14
angl
e (o )
left sloperight slope
to rightunidirectional to left
stoss slope limit
lee slope limit
Figure 7.9. Simulations of dune evolution under reversing winds, but with a dif-fusion coefficient multiplied by 10 from that used to create Figure 7.8. Again, bothslopes adjust to a mean value that depends on the frequency of wind reversals, butthe dune is significantly shallower on both sides. Only the larger dune was used, asthe smaller dune quickly became a symmetric, shallow heap.
this fraction to yield real-time results. As the diffusion coefficient is inversely related
to time (∼ length2/time), this coefficient needs to be increased by the same multi-
plicative scale as time, to accurately reflect the effect of diffusion operating over the
longer real-time period.
Current studies place martian dune evolution on timescales of ten-thousands of
years (Parteli & Herrmann 2007), comparable to estimates for climate shifts and cold
climate processes. This was based on the assumption that the wind speed exceeds the
threshold for saltation initiation only 40s every 5yrs (based on wind speeds recorded
by the Viking spacecraft) – so simulation results were scaled by ∼ 107. Although other
studies claim that saltation should occur more often (Almeida et al. 2008, Kok 2010)
due to the difference between the fluvial threshold and impact threshold for saltation,
the current lack of observed dune migration makes it likely that a temporal scaling
of at least a few orders of magnitude is necessary.
117
Figure 7.10. CO2 frost coats these polar dunes (imaged by HiRISE; 84.7N, 0.8E).As the frost accumulates, it can become translucent ice and cause increased insolationheating at its base. This causes explosive vapor eruptions, which entrain dust andcreate dark spots, rings, and fans at the surface. Such activity can create small chan-nels (Hansen et al. 2007) and mass-wasting events (localized slumps and avalanches)on the dune slope (Horgan et al. 2010). On this dune, there is a clear correlationbetween sublimation spots visible in spring (left) and small avalanche locations visi-ble in summer (right), after the frost has sublimated (examples are highlighted withnumbered markers).
Recent martian dune modeling studies (e.g., Parteli & Herrmann 2007) have not
accounted for this, but have created dune forms that look similar to martian dunes.
This implies that the effective diffusion rate on Mars is comparable to the model
rate, after scaling down by 107 (or whatever time scaling is necessary). Currently
no estimates for diffusion due to stochastic processes exist. One study has estimated
a diffusion rate of 2e-12m2/s for ice-driven creep on martian debris slopes (Perron
et al. 2003). This rate should be small relative to ice-driven creep within granular
material (such as within dunes) and is probably much smaller than diffusion from
aeolian/cold climate processes and avalanching, but provides us with a starting value.
Scaling this diffusion coefficient by 107 yields a simulation diffusion rate of 2e-5m2/s,
which is comparable to the rate used in our simulations for avalanches (Table 3.1).
118
However, as we have shown, model dunes are highly dependent on this simulation
diffusion rate, so it will be crucial to better constrain this value before comparing
simulation results to observed mid-latitude and polar martian dunes.
7.3.1. Possible method for estimating the diffusion rate
The location of the break in a dune’s lee slope may also relate to competition between
saltation and diffusive processes (in addition to reversing winds). Within dunes with
a break in lee slope, the upper portion is the slipface, where saltation is active:
Q ∼ x2/Tsalt, where x is the horizontal length of the slipface (the portion of the lee
slope at the angle of repose; Figure 7.11). Diffusion operates over the entire lee slope:
D ∼ L2/Tdiff, where L is the horizontal length of lee slope. If the dune lee slope
profile is psuedo-steady (i.e, returns to the same value after a wind reversal cycle),
this implies that Tsalt ∼ Tdiff and we find that D/Q ∼ (L/x)2. This relation provides
us with a possible method for estimating the relative diffusion to saltation rate over
the dune based on lee slope morphology, by comparing dunes with similar L/x ratios.
Figure 7.11. Schematic diagram showing lee slope lengthscales that may be relatedto the relative diffusion vs. saltation rate on a dune.
Preliminary simulation results indicate that this may be the case. In these simu-
lations, flux is nearly equivalent between dunes of similar aspect ratio (as the shear
stress calculation depends on nearby-slope values, as discussed in subsection 3.1.2)
and diffusion was a constant. Figure 7.12 shows that the length of the slipface (after
the wind blows towards the right, the dominant direction, and the slope adjusts back)
varies with dune size and frequency of wind reversals. However, the ratio of lee slope
119
to slipface length (L/x) is very similar between simulation runs.
The spread that appears when the period length increases is due to differences
that form between dunes’ aspect ratios, as the dunes are able to more completely
adjust to the new wind direction; this causes differences in the sand flux over the
dunes. This is also the reason that the smallest dune experiencing the 3/1 wind
reversal behaves differently – this dune is able to more completely adjust its shape
between wind reversals, so it has a different shape at the end of a wind reversal cycle
than the other dunes, and does not experience the same flux.
20
30
40
leng
th x
105 106 107 108
2.5
3.5
period length
ratio
L/x
3/1 ratio9/1 ratio
Figure 7.12. Simulations show that the length of the slipface (x) varies with dunesize and frequency of wind reversals, and appears independent of the wind reversalperiod length. The ratio of the lee slope to slipface (L/x) appears independent of allof these values as long as the dune aspect ratios remain similar (not the case with verylong period lengths or very small dunes), so may be primarily related to the relativerates of diffusion and saltation. (Large markers: dunes 30m high; small: 20m.)
7.4. Results
Our simulation results demonstrate specific and quantitative connections between
dune slope morphology and reversing wind directions and cold climate processes (Ta-
120
ble 7.1). Although these results are preliminary and indicate the high-level of cou-
pling between processes, careful combinations of measurables can perhaps be used to
decouple the effects of these processes in the interpretation of actual dune forms. Ad-
ditionally, climate, polar process, and sediment cementation studies will aid efforts by
independently constraining some model parameters, such as effective diffusion rates.
Dune Morphology Related Process/Parameterround/flattened tops diffusion, wind reversal, influx of sand to sidessymmetry of dune/dual slipfaces ratio for wind reversal, cementationslipface length diffusion vs. saltation, period of wind reversalmean lee/stoss slopes ratio for wind reversal, diffusion
Table 7.1. Table showing connected dune slope measurables and the simulatedprocess/parameter associated with such features.
These results will also aid in future efforts to replicate a specific martian dune
shape. If a study successfully replicates the details of a dune form, this implies that
the simulation parameters and processes used are related to and (relatively) scaled
appropriately to the environmental conditions and formative processes of the observed
dune. Although there is no guarantee of this relation, model parameters can provide
at least a starting dataset for understanding a dune’s environment.
This may prove especially useful in understanding the formation of “extreme” dune
shapes, such as flat-topped dune seen in Antartica (Figure 7.13) and on Mars (Figure
7.14). Currently, we have been unsuccessful in numerically replicating this dune
form; it appears that a very carefully selected combination of high diffusion, slope-
dependent ice cementation, and reversing winds will be needed. Another interesting
and currently unexplained dune form is the elongated and slipface-less Antarctic
‘whale-back’ dune (Bristow et al. 2009).
121
Figure 7.13. An Antarctic reversing dune with a flat top, displaying slipface devel-opment on both sides. The west facing ‘summer’ slipface is visible on the right sideof the dune. Image taken from Bristow et al. (2010).
Figure 7.14. HiRISE image of a flat-topped dune with slipfaces on two sides(PSP 004235 1300; 49S, 34E) in Matara crater dune field. The sinuous gullies onthe left of the dune are hypothesized to be formed by melting ground ice (Miyamotoet al. 2004).
122
7.4.1. Implications for martian dunes
Our simulation results show that polar processes and reversing winds cannot be ne-
glected in modeling studies of martian mid-latitude and polar dunes. Such pro-
cesses may explain observed morphological differences between mid-latitude intra-
crater dunes and polar dunes (Bourke, Balme & Zimbelman 2004). Additionally,
steeper mean stoss slopes and shallower mean lee slopes have important implications
for estimations of dune height (e.g., Bourke et al. 2006) and volume, which relate to
estimates of regional and global erosion rates and sediment volume.
123
8. CONCLUSIONS
The aim of this dissertation work was to investigate the role that various environ-
mental conditions and dynamic processes play in determining dune and dune field
morphology. We have approached this through a mixture of model development,
analysis, numerical simulation, and comparison with observations. In Chapter 3, we
constructed and analyzed the dune evolution model. Chapters 4 through 6 focused on
identifying and quantifying the influence that sand flux, dune collisions, and bedrock
topography will play in dune size regulation. In Chapter 7, we focused on details of
dune slope and morphology, and extended the dune evolution model to include re-
versing winds and cold climate processes. Detailed conclusions were provided within
each chapter; here, only general conclusions are given, along with broad discussion of
implications for terrestrial and martian dunes and dune fields. We also briefly discuss
possible future extensions of these studies.
8.1. Dune Field Morphology
In section 1.1, we first discussed the fact that many dune fields contain uniformly-sized
dunes. Many studies have used this to argue that a dune field evolves into specific
patterns and that dunes will tend towards a maximum or equilibrium size, but this
argument has yet not been validated or extended into quantitative connections with
specific environmental conditions or processes. This dissertation aimed to address
this open question.
In Chapter 4, we demonstrated that there is no equilibrium isolated dune size.
Instead, dunes will achieve an unstable equilibrium size based on the incoming sedi-
ment flux; within a field where dunes interact only through the sand flux, this leads
to coalescent behavior.
124
This result motivated the focus of Chapter 5, which was to investigate the influ-
ence that interacting dunes will have on dune size within a field. We first considered
the influence that non-flat topography (such as neighboring dunes) will have on shear
stress, and thus on sand flux. This also predicted coalescent dynamics, so we focused
instead on mass exchange through dune collisions. This study is not the first to
construct a dune field model, but it is the first to sequentially and thoroughly ex-
plore the components of contemporary dune field models and to identify the crucial
elements that are needed to connect model results to an observed dune field. We
identified specific criteria for a patterned dune field to form, and quantitatively con-
nected the dune field’s end-state (patterned: Figure 1.2 or runaway growth: Figure
1.3; subsection 5.3.4) with its collision dynamics and influx size population.
Although we showed it is possible for a dune field to become patterned, it was
unclear how likely it is that so many dune fields just naturally have the correct cou-
pling between the dune interaction processes and the dune nucleation size population.
In Chapter 6, we completed preliminary numerical experiments to explore the effect
non-flat bedrock topography would have on isolated dune evolution, and extended
this to binary dune collisions. We had hoped that topography would exert a stabi-
lizing influence and would create more similarly-sized dunes, but it in fact did the
opposite (at least in two dimensions). Both Gaussian hills and upward/downward
steps enhanced interaction between colliding dunes, resulting in a higher sand trans-
fer from the smaller to the larger dune. This increased the probability for coalescence
and overall pushed a dune field towards runaway dynamics.
In summary, we have primarily negative results with respect to dune size regulation
through isolated dune evolution via sand flux and migration over topography. This
strongly implies that studies focused on the evolution of one or a few dunes will not
be sufficient for identifying the reason that dunes within a field often tend towards a
uniform size. In other words, a dune evolution model cannot simply be scaled up for
studies of dune field evolution.
125
Instead, studies need to be multi-scale and to focus on interactions that occur
within the field between neighboring dunes and between dunes and (possibly non-
constant) environmental conditions. For example, in Chapter 5 we showed that dune
collision dynamics can result in the dune field evolving into uniformly-sized dunes,
given a proper coupling between the influx dune size population and the interaction
rule. Further and more detailed investigation is needed here to explore the premise
that most dune fields naturally have the right conditions for pattern formation.
There are other possible processes that may enhance dune field stabilization
through size-dependent effects (and preferential break-up or stabilization of large
dunes). Some of these will be discussed in Section 8.3.
8.2. Dune Morphology
Although isolated dune modeling studies may not be sufficient for understanding dune
field evolution, the dune evolution model has been shown to be useful in analysis of
detailed dune morphology. Past studies using a continuum dune evolution model have
been used to quantitatively connect observed martian and terrestrial dune shapes with
environmental conditions throughout dune evolution.
However, the contemporary dune evolution model is limited as it relies on several
simplifying assumptions about constancy in the environment (such as steady winds
and flat bedrock topography) and dynamics. In this dissertation, we have attempted
to extend the model to accomodate more complicated environments.
In Chapter 6, we showed that non-flat bedrock topography (such as a Gaussian hill
or step) will temporarily alter a dune’s shape and migration rate. Although the dunes
did adjust back to their original shapes and migration velocity (unless the topography
and dune were of comparable height), these changes need to be considered when
making comparisons between simulation results and observed dunes evolving over non-
flat terrain (e.g., in a crater). Alternatively, an understanding of the influence that
126
topography has on dune evolution can perhaps be used to reverse-engineer topography
estimates based on observed dune shapes.
The dune evolution model was also extended to include reversing winds and po-
lar processes, as presented in Chapter 7. Although this results of this study are
ill-constrained due to a lack of comparative observations, our simulation results show
specific and quantitative connections between dune slope morphology and reversing
wind directions and polar processes that could be used in the interpretation of ob-
served polar dune fields on the Earth and Mars. Our results are very preliminary,
but they do provide a framework for future studies aimed at quantifying the influence
polar processes and reversing winds exert on terrestrial and martian dune evolution.
That study also highlights the importance of properly scaling all model parameters
in a consistent manner to accurately portray the effect of interacting processes (such
as gravity-driven avalanches, general diffusion processes, and wind-driven saltation).
This last point will be discussed further in subsection 8.3.3.
8.3. Future Extensions
Here we propose several research extensions that should be included in the construc-
tion of future dune and dune field evolution model studies. Existing observations and
qualitative models imply that these processes exert measurable effects on dune and
dune field morphology, and thus should be considered when interpreting observed
dune forms.
8.3.1. Dune nucleation
In subsection 5.4.1, we showed that two model parameters completely predicted
whether or not a dune field would become patterned after a large number of dune
collisions (Figure 5.12): (i) an interaction parameter describing when collisions will
redistribute sand from the larger to the smaller dune (the crossover value, r∗), thus
127
decreasing dune size polydispersity, and (ii) the the standard deviation/mean ra-
tio of the influx dunes’ size distribution, which corresponds to the polydispersity of
the dune sizes found at the beginning of the field, when dunes first form and have
not yet interacted with each other. Previous studies have calculated dune size and
spacing distributions within mature terrestrial (Wilkins & Ford 2007) and martian
(Bishop 2007) dune fields and proved that dunes undergo some form of organization to
create non-random distributions. These studies, however, did not explain the origin
of the organization, which could depend on processes and environmental conditions
during dune formation and/or subsequent interaction.
A method for quantifying the size distribution created through dune nucleation
within observed fields would provide constrained estimates for the influx population
used in a dune field model. This will allow modeled dune fields to more closely resem-
ble specific, observed dune fields and increase the prediction ability of our dune field
model. Additionally, by isolating the effect of formation processes and conditions on
dune field morphology, this study will provide much-needed validation and calibration
information for dune initiation models.
8.3.2. Evolving intrafield sediment conditions
Currently, the dune evolution model considers only the mean grain size within the
dune, and even this is indirectly included within a flow parameter that directly relates
to the minimum dune size (Andreotti et al. 2002b) and dune destabilization (Elbelrhiti
et al. 2005): the sand flux saturation length (a spatial delay in the sand flux’s response
to the local wind regime). As this flow parameter is generally held constant in dune
evolution model studies, it is implicitly assumed that the mean grain size does not
greatly change throughout a simulation.
Field studies have shown that dunes may nucleate with a strongly peaked and
spatially-uniform grain distribution, but spatial variations in mean grain size and
128
grain size distribution naturally evolve (Figure 2.3) as a function of wind regime, dune
age/type, and underlying sediment composition (Wang et al. 2003, Besler 2005). As
briefly mentioned in subsection 5.4.4, increasing observational evidence implies that
these variations can strongly influence dune and dune field dynamics. For example,
Besler (2002) examined apparent collision dynamics within on-going dune collisions in
the Libyan desert, and compared this with the dunes’ granulometrics. Based on these
observations, she hypothesized that dunes made of softer and finer grains were more
likely to coalesce, while interacting dunes made of more compacted and coarser grains
was more likely to have a collision result in separate, similarly-sized dunes (enhancing
stability of a patterned dune field). Due to winnowing effect, older/larger dunes are
more likely to contain more compacted, coarser grains (Besler 2005), meaning that
dune collision dynamics are not constant, and should enhance pattern formation as
a dune field ages. Additionally, as surface grains increase in mean size, sediment
transport rates should slow and older dunes should be stabilized.
This stabilization of more mature dune fields due to granule surface shielding,
along with the overall generally larger size of older dunes, can perhaps explain why
dune field reorganization seems to occur over much slower timescales than dune
construction, leading to multiple dune patterns superimposed within a dune field
(Kocurek & Ewing 2005).
8.3.3. Process interactions
As discussed in Chapter 7, when multiple processes are relevant in dune evolution,
it is important to estimate (or at least constrain) the relative timing and param-
eter magnitudes related to the different processes. This is especially important if
the processes are interacting to create the dune form; for example, it is competition
between diffusive processes and wind-driven saltation that create the stoss/lee dune
slopes and the asymmetry in the dune cross-sectional profile (Figure 2.1, 2.2; Sec-
129
tion 2.2). Estimates of these relative process rates can sometimes be found through
independent studies, thus allowing for predictions of dune evolution and morphol-
ogy. Conversely, the relative rates can sometimes be estimated from observations and
scaling arguments (e.g., subsections 3.1.5, 7.3.1).
It is also important to investigate how discrete processes will interact with con-
tinuous or averaged processes. For example, in Chapter 7 we showed that discrete
reversals of wind direction influence some dune measurables (such as an instantaneous
measure of the dune slope), while the time-averaged process (which can be approx-
imated as a continuous process) influenced other measurables (such as the average
dune stoss slope; Figure 7.7).
This is also important when a process occurs over much longer timescales (such as
ice-cementation of sediment in polar regions) than changes in mobile-sediment supply
and or wind conditions. As pointed out in Chapter 7, studies using discrete intervals
of saltation (rather than a temporally-scaled continuous-saltation simulation) may be
especially important in martian dune studies, as saltation is thought to occur only
rarely (Parteli & Herrmann 2007) and may interact with slower, but more continuous,
cold climate processes.
Finally, it has also been proposed that seasonal storms play an important role
in regulating dunes’ sizes by inducing long-wavelength perturbations on large dunes,
which cause them to break into several dunes (Elbelrhiti et al. 2005). Given the
observational support for this theory, it would be very useful to include this stochastic
process in the dune field model. However, this will first require a quantification of
storm occurrence rates, and exploration of how important the discrete events will be,
versus considering only the long-time-averaged influence.
130
8.4. The Need for Comparative Observations
As discussed in Chapter 1, increased collaboration between dune geologists, physical
modelers, and mathematicians has greatly aided efforts to quantitatively connect
dune and dune field evolution with environmental conditions and physical processes.
For example, the increased availability of spacecraft images of dune fields on different
planetary surfaces in the last decade has helped with the validation of assumptions
and parameters used in the basic dune evolution model. Currently, however, the
main constraint on further dune and dune field evolution model development remains
a lack of field or laboratory observations. In particular, there are very few studies
which focus on dune interactions and nucleation, which greatly limits model predictive
ability.
Studies such as those presented in this dissertation can, however, serve as as
guides for future observational studies. Scaling relations and behavior trends derived
through model equation analysis and numerical experimentation provide information
about possible connections between dune measurables and influential environmental
conditions or physical processes. For example, this dissertation highlights the need
for these field and laboratory studies:
• Changes in localized saltation rates as dunes collide with a neighboring dune
(Section 5.2) or with bedrock topography (Section 6.2);
• Measurements of relative saltation and diffusion rates, and resultant dune stoss
and lee slopes (Section 7.3);
• Dune size distribution found at the beginning of the dune field and created
through nucleation processes (subsection 8.3.1).
Future interative comparisons between mathematical models of dune and dune
field evolution and field and laboratory observations will continue to elucidate, in
131
greater and greater detail, the quantitative connections between environmental con-
ditions and dune morphology/behavior. As dunes are prime examples of geomorphic
markers of environmental conditions over many scales, these studies provide impor-
tant contributions towards the advancement of scientific knowledge and understand-
ing of surface and climate evolution on the Earth and other planets.
132
A. THE NUMERICAL ALGORITHM
As outlined in subsection 3.1.8, the simulation evolves the non-dimensional dune
profile h over a time step dt, via several relations. Within our model, all relations
have been linearized to simplify analysis.
Within each time step, we start with the dune profile (hn). The following steps are
then followed to yield the dune profile at the next time step (hn+1), and the process
is iterated.
A.1. Avalanching
Avalanching is applied to smooth the surface and to keep slopes below the angle of
repose. This process is assumed to be diffusive in nature, as described in subsection
3.1.5. The diffusion coefficient at a mid-gridpoint (xi−1/2, which is between xi and
xi−1; i ∈ [1, 2, . . . , N ]) is calculated via backward differences:
D(xi−1/2) = Dh
1 + Da exp((|(hi − hi−1)/dx| − tan(ref. angle))/γ
)1 + exp
((|(hi − hi−1)/dx| − tan(ref. angle))/γ
)︸ ︷︷ ︸slope check
×
(1
2tanh(G1(hi − gri −G2ε)) + 1/2
)︸ ︷︷ ︸
ground check
(A.1)
The leading coefficient (Dh) is the low-level diffusion that smooths the surface due to
wind variability, etc. The ‘slope check’ term (Figure A.1) determines if avalanching
should occur; if so, then the diffusion coefficient is increased by the multiplicative
factor Da (generally 102 or 103). The ‘ground check’ term (Figure A.2) determines
how much sand is available for avalanching; no avalanching occurs when within G2ε
(the limit ε is defined in subsection 2.2.3) of the the non-erodible surface (gr), which
was generally defined as h = 0 (sometimes more complicated topographies were used,
133
as described in Chapter 6). When periodic boundary conditions were used, then
h0 = hn; when semi-infinite boundary conditions were used, then D(x1/2) = 0 to fix
the end-gridpoint.
20 30 40 50 600
20
40
60
80
100
slope (o)
slop
e ch
eck
1
100 (Da)
Figure A.1. Function used to determine diffusion coefficient, based on local slope.When the slope is below the angle of repose (vertical dashed line), then this function→ 1. When the slope is at or above the angle of repose, then this function → Da.The exact shape of this function is determined by the ref. angle (which determineswhen it increases; generally ref. angle = 40o) and γ (which determines its steepness;γ = 0.02).
An implicit central differences scheme was then applied to calculate diffusion, with
a forward difference in time:
ht = (D(hx)hx)x
hn+1i − hn
i
dt=
Di+1/2hi+1 − (Di+1/2 + Di−1/2)hi + Di−1/2hi−1
dx2(A.2)
A.2. Separation bubble
We then calculate the separation bubble (s) which describes the airflow over the
surface. This is done by locating all brink locations, and then extending the upper-
right portion of an ellipse (of β = 6.5 aspect ratio, as defined in subsection 3.1.1)
from each brink, with continuous value and slope at the connection point. The
134
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
(h−gr)/ε
grou
nd c
heck
1
0
Figure A.2. Function used to determine diffusion coefficient, based on availabilityof mobile sand. When the sand depth is less than G2ε, then avalanching does notoccur. The exact shape of this function is determined by the G1 (which determinesthe steepness; here G1 = 200) and G2 (which determines at what depth the sandshould not avalanche; G2 = 5).
airflow profile for the entire simulation box is then found by taking the maximum of
the juxtaposition of these partial-ellipse-profiles and the dune topography. The full
profile is C1 except where a separation bubble rejoins downwind topography (where
it is C0).
A.3. Shear stress
The shear stress perturbation (τ ) is calculated from the separation bubble (s) via
fourier methods (Matlab FFT) applied to the Jackson-Hunt equation (Jackson &
Hunt 1975, Weng et al. 1991):
τ = (A ·K + B ·D) · s (A.3)
where K is the the convolution of 1/πx (the Hilbert transform) and the spatial-
derivative in fourier space (|k|), and D is the spatial-derivative in fourier space (ik).
The parameters A and B determine the importance of the local versus global slope,
135
and depend logarithmically on the ratio of geometric lengthscale (conventionally, the
dune half-length; L = π/2k) to the surface roughness (∼ 110
grain diameter; z0). For
a dune, the geometric lengthscale is typically 6-9 orders of magnitude longer than
the surface roughness, so A ∼ 4 and B ∼ 1. These parameters can be calculated
more exactly via the following equations (explained in more detail in Jackson &
Hunt 1975, Weng et al. 1991, Kroy et al. 2002):
Φ ≡ L/z0
φ ≡ 2κ2Φ/ ln φ (note implicit formula) (A.4)
A =ln(Φ2/ ln(Φ))2
2 ln(φ)3
(1 + ln(φ) + 2 ln(π/2L) + 4γE
)(A.5)
B = πln(Φ2/ ln(Φ))2
2 ln(φ)3(A.6)
where κ is the von Karman constant (≈ 0.4) and γE is Euler’s constant (≈ 0.577)
A.3.1. Jackson-Hunt equation
In this subsection, we briefly describe the derivation of the Jackson-Hunt equation
(which is explained in full detail in Jackson & Hunt 1975, Weng et al. 1991). We focus
on this equation as its inclusion is perhaps the weakest assumption within our dune
evolution model. The use of this “shallow hill” approximation has been empirically
validated (Sauermann et al. 2003) for flow over an isolated dune on a flat surface, but
it may (is probably) not be a valid approximation over more complicated terrain and
dune morphologies. However, at this time no better approximation is known.
The study by Jackson & Hunt (1975) considers windflow over a two-dimensional
low, symmetric hill (Figure A.3; H � L and slope everywhere on the order of H/L,
the study used h(x/L) = H/(1+(x/L)2)) with constant surface roughness (z0; assume
L/z0 → ∞; note that L/z0 = Φ as in Eqn. A.4) and far-field shear stress velocity
(u∗). Far above and upwind/downwind from the hill (i.e., as y/L or |x/L| → ∞), the
136
airflow will follow the Prandtl-von Karman model (Eqn. 2.1):
u0(y) =
{u∗/κ ln(y/z0) y < δ
u∗/κ ln(δ/z0) y ≥ δ (free stream velocity).(A.7)
u (δ)0
u (y)δ
linner region
outer region
free stream flow
y
xL
Hhill profile = h(x/L)
0
Figure A.3. Schematic diagram showing airflow regimes over a low hill. Image isfrom Jackson & Hunt (1975).
Over the hill, the study considers flow within two layers. Within the inner layer
(of depth �; note that �/z0 = φ from Eqn. A.4), the horizontal velocity (u) is given
to the first approximation by the upstream velocity at the same displacement above
level ground (Δy = y−h(x/L); y is the height over the far-field flat plane). However,
continuity implies the existence of a vertical velocity (due to airflow compression
from the hill: v = hxu0(Δy)) which causes a perturbative pressure (p) on the outer
region, and thus a change in horizontal velocity within the outer region (u(x, y) =
u0(y) + u(x, y)). Continuity of pressure between the regions then implies that there
is also a horizontal perturbation velocity within the inner region (i.e, within the
inner boundary, the wind velocity at a local height over the hill is that given by the
Prandtl-von Karman model plus a perturbation: ui(x, Δy) = u0(Δy) + ui(x, Δy).
These perturbations (u and v) are first calculated within the outer region. They
are further constrained (and ui is calculated) by matching velocity and pressure terms
137
at the boundary between the inner and outer layers.
Solving for the flow within the outer region involves consideration of the vertical
displacement of the flow due to presence of the hill (in the inner region). The outer
flow’s upper and far-field boundary conditions are set by the far-field conditions, as
far from the hill (i.e., as y/L or |x/L| → ∞) the displacement and thus the vertical
velocity u should tend to zero. This suggests that the vertical scale of the outer region
(δ) must be of the same order as the horizontal scale of the inner region, which is the
length of the hill L.
The perturbation terms within this layer (u, v, and p) can be expressed in asymp-
totic power series for the limit ln Φ →∞:
i.e., u ∼ u∗ln φ
ln Φ(U (0) +
1
ln ΦU (1) + . . . ) (A.8)
Normalizing u so that the linear inertial terms are of the same order as the pressure
gradient terms, then substituting into the momentum equation (up to O(1/ lnΦ))
yields, for y < δ:
∂U (0)
∂(x/L)+
∂P (0)
∂(x/L)= 0 (A.9)
∂V (0)
∂(x/L)+
∂P (0)
∂(x/L)= 0 (A.10)
∂U (1)
∂(x/L)+
∂P (1)
∂(x/L)= −
(ln(y/L)
∂U (0)
∂(x/L)+
∂V (0)
∂(x/L)
)(A.11)
∂V (1)
∂(x/L)+
∂P (1)
∂(x/L)= − ln(y/L)
∂V (0)
∂(x/L)(A.12)
(A.13)
This yields:
∇2V (0) = 0 (A.14)
∇2V (1) = V (0)/(y/L)2 (A.15)
where ∇2 = ∂2
∂(x/L)2+ ∂2
∂(y/L)2.
138
For y > δ, as du0/dy = 0 the equations lead to:
∇2V (0) = 0 (A.16)
∇2V (1) = 0
To justify the neglect of higher order terms, the following scale constraints are
needed: ln Φ ln(δ/L) and ln(δ/L) = O(1). As long as these conditions are satisfied,
then the asymptotic expansion (Eqn. A.8) will be valid within the outer region for
both y < δ and y > δ.
At the boundary between the inner region and the outer region (i.e., as y/L→ 0
in the outer region and Δy/� → ∞ in the inner region), the perturbative pressure
(p) and horizontal and vertical velocities (ui, v) must match. Solving Eqn. A.12 and
A.16 in Fourier space (w.r.t. the x-coordinate) with all boundary conditions yields:
P (0)(x) = − L
πH
∫∞
-∞
hx(s)
(x/L− s)ds (A.17)
on the surface y/L → 0. The lower boundary of the inner region (δy = z0) must
satisfy the no-slip condition (i.e., u = v = 0). The thickness of the inner region (�) is
found by considering the necessary balance between the acceleration and the stress
gradient when Δy/� ∼ O(1).
As was done for the outer region, the velocities and pressure can be expanded
asymptotically within a power series for the limit ln(φ)→∞:
i.e., ui ∼ U(0)i +
1
ln φU
(1)i + . . . (A.18)
139
which yields:
∂U(0)i
∂(x/L)+
∂P(0)i
∂(x/L)=
∂
∂(Δy/�)
(Δy
�
∂U(0)i
∂(Δy/�)
)(A.19)
∂U(1)i
∂(x/L)+
∂P(1)i
∂(x/L)− ∂
∂(Δy/�)
(Δy
�
∂U(1)i
∂(Δy/�)
)= (A.20)
− ln(y/�)∂U
(0)i
∂(x/L)− V
(0)i
y/�+ 2a1κ
2 ∂
∂(Δx/L)
(Δy
�
∂U(0)i
∂(Δy/�)
)∂P
(0)i
∂(Δy/�)= 0 (A.21)
∂P(1)i
∂(Δy/�)= 2a2κ
2 ∂
∂(Δy/L)
(Δy
�
∂U(0)i
∂(Δy/�)
)(A.22)
Solving these equations yields:
u ∼ u∗
(ln(y/z0)− h
y− H
Lln(Φ)P (0) + . . .
)(A.23)
which, as expected, is the asymptotic solution at both the lower bound of the outer
region (y/L → 0) and the upper bound of the inner region (Δy/� → ∞). The first
term in Eqn. A.23 is u0(y) (Eqn. A.7), so the remainder is the perturbative horizontal
velocity (u: Figure A.4).
u(Δy)~
x/L
y/L
1
1-1 0
Figure A.4. Normalized perturbative velocities (u(x, Δy)) at different positions atthe surface of a low hill. Image is from Jackson & Hunt (1975).
140
Once the perturbative horizontal velocity is calculated, it can be rewritten as a
perturbation to the shear stress (τ = (u∗)2ρair):
τ ∼ 2κΔy
u∗
∂u
∂y(A.24)
A.4. Sand flux
As we are working in the non-dimensional space (i.e., have normalized by the far-field
sand flux Cτ3/20 ), the saturated sand flux qs = 1 + 3/2τ .
The actual sand flux is found via backward differences:
qx = qs − q
q(xi)− q(xi−1)/dx =
⎧⎪⎨⎪⎩qs(xi)− q(xi) hi − gri > ε and h = s
0 hi − gri < ε and h = s
−q(xi) h < s
(A.25)
Within the second case, sand is neither deposited nor eroded as bedrock is exposed.
Within the third case (s > h only inside the shadow zone/separation bubble), qs ≡ 0
so only deposition occurs. These two cases preserve the nonnegativity of h, as qx ≤ 0
(only deposition can occur) as long as h is small.
When periodic boundary conditions are used, q is calculated via matrix inversion.
When a semi-infinite boundary conditions are used, then q is calculated from a fixed
q1 = qin.
A.5. Update the dune profile
Changes in dune profile are then found via the Exner equation, discretized via up-
winding (backward differencing, as sand blows in one direction):
ht = −qx
(hn+1i − hn
i )/dt = −(qi − qi−1)/dx (A.26)
141
B. TWO-DUNE REDUCED DIMENSION MODEL
This model was constructed to investigate the dynamics of a collision between two
dunes within a reduced complexity frame. The aim was to isolate important behaviors
without consideration of detailed dune morphology. Unfortunately, although this
model can be used to simulate binary dune collisions, we were unable to decouple
the evolution of the six model variables, so the reduced dimension system was not
actually easier to simulate or analyze.
B.1. Two-dune Structure
We specify the total cross-sectional area (A) of the two dunes, and assume that it
is conserved throughout the collision. The lee slopes are fixed at the slope of repose
(α = tan(34o)), and the dunes are initiated in contact with each other. The two-
dune profile (h; Figure B.1) is now specified by five variables: the windward slopes
of the dunes (σu,d, where ‘u’ denotes upwind and ‘d’ denotes downwind), the heights
of the dunes’ crests (Hu,d), and the distance between the foot of the structure and an
arbitrary location (m). The distance between the dunes’ crests (d) is fixed since we
specified A, and is equal to:
d =H2
σd+
Hu
α−
√2Ao
1/α + 1/σd
( 1
α+
1
σd
)where Ao is the “overlap” volume:
Ao =H2
u
2
( 1
σu
+1
α
)+
H2d
2
( 1
σd
+1
α
)− A.
Note that d is positive (and physically meaningful) only if A0 ∈ [0, A]. Addition-
ally, if A0 = 0 (i.e., the two interacting dunes are disjoint), then d is not uniquely
142
���
�
�� ���� ��
�
�
��
��
!"# !$&'*
*
Figure B.1. Schematic diagram of the reduced dimension two-dune structure.
defined (d ≥ Hd
σd+ Hu
α); to uniquely define d, we restrict the dunes to be just touching
(d = Hd
σd+ Hu
α) .
We define the dune profile (h(x, t); Figure B.1), which is piece-wise linear and
continuous with six regions:
h =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 x ∈ PRE = (−∞, m), pre-dune
(x−m)σu x ∈Wu = (m, m + Hu
σu),
upwind dune’s stoss slope
−(x−m− Hu
σu)α x ∈ Lu = (m + Hu
σu, m + Hu
σu+ Hu−Hd+dσd
σd+α),
+Hu upwind dune’s lee slope
(x−m− Hu
σu− d)σd x ∈Wd = (m + Hu
σu+ Hu−Hd+dσd
σd+α, m + Hu
σu+ d),
+Hd downwind dune’s stoss slope
−(x−m−Hu/σu − d)α x ∈ Ld = (m + Hu
σu+ d, m + Hu
σu+ d + Hd
α),
+Hd downwind dune’s lee slope
0 x ∈ POST = (m + Hu
σu+ d + Hd
α,∞),
post-dune
(B.1)
143
B.2. Derivatives of the Two-dune Profile
Derivatives of this function with respect to the variables are as follows:
∂h
∂σu
=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 x ∈ PRE
(x−m) x ∈Wu
−Huασ2
ux ∈ Lu
Huσd
σ2u
(1−Hu/2
√2Ao
1/σd+1/α
)x ∈Wd
−Huασ2
u
(1−Hu/2
√2Ao
1/σd+1/α
)x ∈ Ld
0 x ∈ POST
(B.2)
∂h
∂σd
=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 x ∈ PRE
0 x ∈Wu
0 x ∈ Lu
x−m−Hu/σu − d
− 12σ2
d
(H2
d/√
2Ao
1/σd+1/α+−
√2Ao
1/σd+1/α+ 2Hd
)x ∈Wd
− α2σ2
d
(H2
d/√
2Ao
1/σd+1/α−
√2Ao
1/σd+1/α+ 2Hd
)x ∈ Ld
0 x ∈ POST
(B.3)
∂h
∂Hu=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 x ∈ PRE
0 x ∈ Wu
ασu
+ 1 x ∈ Lu
−σd
(1
σu+ 1
α−Hu(1/σu + 1/α)/
√2Ao
1/σd+1/α
)x ∈ Wd
α(
1σu
+ 1α−Hu(1/σu + 1/α)/
√2Ao
1/σd+1/α
)x ∈ Ld
0 x ∈ POST
(B.4)
∂h
∂Hd=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 x ∈ PRE
0 x ∈Wu
0 x ∈ Lu
Hd(1 + σd/α)/√
2Ao
1/σd+1/αx ∈Wd
−Hd(1 + α/σd)/√
2Ao
1/σd+1/α+ α/σd + 1 x ∈ Ld
0 x ∈ POST
(B.5)
144
∂h
∂m=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 x ∈ PRE
−σu x ∈Wu
α x ∈ Lu
−σd x ∈Wd
α x ∈ Ld
0 x ∈ POST
(B.6)
0 5 10 15 20 25
−100
−50
0
50
100
σσHH
u
ud
d
uW uL dW dL
Figure B.2. Plot of numerically computed partial derivatives of the dune profile(h), with respect to Hu, Hd, σu, and σd in regions Wu, Lu, Wd, and Ld.
Assuming that h can be expressed purely as a function of the five variables
(σu, σd, Hu, Hd, m), we have:
dh
dt=
∂h
∂σuσu +
∂h
∂σdσd +
∂h
∂HuHu +
∂h
∂HdHd +
∂h
∂mm, (B.7)
where () ≡ ∂h∂t
. Projecting this function onto a space spanned by the partial derivatives
of h yields the time-derivatives of the different variables.
145
B.3. Isolating Time-derivatives
We identify a set of functions that spans the same space as the partial derivatives
Figure B.2, as follows:
• Test function one (T1) is a linear function on the windward slope of the first
dune, and isolates σu:
y =
{x−m− Hu
2σux ∈Wu
0 otherwise
• Test function two (T2) is a linear function on the windward slope of the second
dune, and isolates σd:
y =
{x−m− Hu
σu− Hd−Hu+αd
2(σd+α)x ∈Wd
0 otherwise
• Test function three (T3) is a constant function over the windward slope of the
first dune, and it isolates m:
y =
{1 x ∈Wu
0 otherwise
• Test function four (T4) is a constant function on the leeward slope of the first
dune, and isolates (after subtracting off known parts from the above derivatines)
Hu:
y =
{1 x ∈ Lu
0 otherwise
• Test function five (T5) is a constant function on the leeward slope of the second
dune, and isolates (after subtracting off known parts from the above derivatines)
Hd:
y =
{1 x ∈ Ld
0 otherwise
146
We now can use the inner product:
〈f, g〉 =
∫∞
−∞
f(x)g(x)dx
to project the RHS of the conservation equation (ht = −qx) onto the space spanned
by the test functions.
Test function one gives us:
〈T1,−qx〉 = 〈T1,∂h
∂σu
σu〉
=
∫Wu
(x−m− Hu
2σu
)(x−m)σudx
=H3
u
12σ3u
σu
⇒ σu =12σ3
u
H3u
〈T1,−qx〉 (B.8)
which is the result found with the one-dune reduced model (Equation 3.26). The
two-dune model also reproduces the results of the one-dune model for σu (Figure
B.3) when run with only one dune (i.e., let Hd = 0; one-dune model is discussed in
subsection 3.2.2).
Similarly, test function two yields:
σd =12σ3
d
(Hd −√
2Ao
1/α+1/σd)3〈T2,−qx〉. (B.9)
Test function three yields:
〈T3,−qx〉 =
∫Wu
( ∂h
∂σu
σu +∂h
∂mm
)dx
⇒ m =1
Hu
(− 〈T3,−qx〉+ 6
σu
Hu
〈T1,−qx〉). (B.10)
Note that if we have a steady windward slope on the first dune (σu = 0), then we
would just have:
m =−〈T3,−qx〉
Hu=
qonto first dune’s foot − qout over first dune’s crest
Hu
147
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−6
−4
−2
0
2
4
6x 10−3
σ
dσ/d
t
Figure B.3. Dynamics of windward slope as a function of slope for several differentsize dunes, computed using the two-dune model. Results have the same general shapeand characteristics as those found with the one-dune model: Figure 3.6. Some noiseis present due to small transitive variations in the dune profile.
which is the same as that found with the one-dune model (Equation 3.28).
Test function four yields:
〈T4,−qx〉 =
∫Lu
( ∂h
∂σuσu +
∂h
∂HuHu +
∂h
∂mm
)dx
⇒ Hu =( 〈T4,−qx〉
Hu −√
2Ao
1/α+1/σd
+6σu〈T1,−qx〉
H2u
+〈T3,−qx〉
Hu
)/( 1
α+
1
σu
)(B.11)
148
Finally, test function five yields:
〈T5,−qx〉 =
∫Ld
( ∂h
∂σuσu +
∂h
∂σdσd +
∂h
∂HuHu +
∂h
∂HdHd +
∂h
∂mm
)dx
⇒ Hd =( α
Hd〈T5,−qx〉 − ∂h
∂σuσu − ∂h
∂σdσd − ∂h
∂HuHu − ∂h
∂mm
)σd ÷(
(σd + α)(1−Hd/
√2Ao
1/α + 1/σd))
=(〈T5,−qx〉/Hd +
12σu
H2u
(1− Hu
2√
.
)〈T1,−qx〉+
6σ2d
(Hd −√.)3
(H2d√.−√. + 2Hd
)〈T2,−qx〉
−(1− Hu√
.
)(〈T4,−qx〉Hu −√.
+6σu〈T1,−qx〉
H2u
+〈T3,−qx〉
Hu
)−
σd
αHu
(6σu〈T1,−qx〉Hu
− 〈T3,−qx〉))÷((
1− Hd√.
)Hd(σd + α)
ασd
)(B.12)
With this reduced dimension simulation of binary dune collisions, coalescence
occurred when the downwind dune’s crest dropped onto the upwind dune’s lee slope.
If this did not happen (Hd remained higher than h ∈ Lu), then ejection resulted.
Tests showed that predicted dynamics were consistent with those found with the full
dune evolution model (Section 5.2), but the reduced dimension simulation returned
less information with regards to the ejected dune’s final size. The simulation also was
not simpler in use or faster, so this approach was abandoned.
149
C. SMOLUCHOWSKI DUNE INTERACTION STUDY
Smoluchowski coagulation equations are used to to describe the evolution of the
number density P of particles of size x at a time t. This is useful when systems consist
of a very large number of particles, such as within gaseous molecular interactions.
In the continuous case, the Smoluchowski equation is:
∂P (x, t)
∂t=
1
2
∫ x
0
K(x− y, y)P (x− y, t)P (y, t)dy−∫∞
0
K(x, y)P (x, t)P (y, t)dy,
where the operator (K) is known as the coagulation kernel and describes the rate
at which particles of size x interact with particles of size y. The first term on the
RHS of the equation corresponds to interactions which increase P(x), and the second
corresponds to interactions which remove elements from P(x).
If we were able to derive the proper form of the kernel function for a dune field,
we could perhaps use this framework to analyze the evolution of the number density
of dunes of size m within the field.
Unfortunately, the dune field system differed from the physical systems analyzed
by Smoluchowski in two key ways: interactions are asymmetric, as dunes and sediment
move only downwind; secondly, dunes do not coagulate, but instead exchange mass
when they collide.
Although this approach was abandoned in favor of a discrete particle model (dis-
cussed in sections 5.3), here we outline our kernel function derivation. The kernel
function is a compilation of two functions: the rate (R) at which an upwind dune of
mass abefore and a downwind dune of mass Abefore interact, and the probability (O)
that this interaction will yield a dune of size M .
150
C.1. Rate Function
As we have fixed a scale-invariant dune morphology, it is irrelevant if we consider dune
height or area; as dune velocity is related to height, for simplicity the following relation
will be given in terms of height (and height ∼ area1/2). We consider interacting dunes
of pre-collision height h and H . As dunes move in one direction, following the wind
direction, R(h, H) = 0 if the upwind dune (the first argument) is smaller than the
downwind dune. Assuming h < H, their rate of interaction will be determined by
their relative velocity. In the frame of reference of the slower dune, the two dunes will
interact once per time needed for the faster dune to do one orbit of the simulation
box (if periodic boundary conditions are assumed). Thus, the rate of interaction is
R = 1/T , where T is the time length of this orbit. As we are in the reference frame of
the slower dune, the velocity that the faster dune is moving at is v− V = h−1−H−1
(= a−1/2 − A−1/2). The distance traveled is the length of the simulation box (L), so
T = Lv−V
, or:
R =h−1 −H−1
L=
H − h
LhH(C.1)
which is illustrated in Figure C.1.
To validate this analytical result, we counted dune collisions within randomly
generated dune fields (Figure C.2). Note that this relationship is quite different
from the rate function found for collisions between gas molecules (which exchange
velocities), where R ∼ 1h2H2 .
C.2. Output Function
Our output function will be in the form of a δ-function, as it will be determined based
on numerical simulation of our full dune evolution model, which is deterministic. Our
output function must also be a superposition of two cases, as sometimes one dune
is the result of interaction (coalescence) and sometimes two dunes result (ejection).
151
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
0
1
2
3
4
5
6
7
8
9x 10
−3
a
A
Figure C.1. Plot of the rate of interaction (number of collisions per unit time)between an upwind dune of size a ∼ h2 and a downwind dune of size A ∼ H2
(equation C.1).
We define the following case-functions (where the a and A again refer only to the
pre-collision sizes, and the first argument refers to the upwind dune size):
Oc(a, A, M) = Cc(a, A)δ(a + A−M) (C.2)
where Cc(a, A) =
{1 coalescence occurs
0 ejection occurs(C.3)
Oe(a, A, M) = Ce(a, A)(δ(Mafter, upwind(abefore, Abefore)−M) + (C.4)
δ(Mafter, downwind(a, A)−M))
(C.5)
where Ce(x, y) =
{0 coalescence occurs
1 ejection occurs. (C.6)
The full output function is, thus, O(a, A, M) = Oc(a, A, M) + Oe(a, A, M). For
any given a and A, the integral of O over all output dune sizes (M) yields the total
number of dunes which result from the collision:∫∞
0
O(a, A, M)dM =
{1 coalescence occurs
2 ejection occurs. (C.7)
To determine the exact form of the output function, we need to know how the
152
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
0
1
2
3
4
5
6x 10
−3
a
A
Figure C.2. The number of interactions (number of collisions per unit time) betweenan upwind dune of size a and and a downwind dune of size A, as counted within 30simulation runs.
masses of two interacting dunes influences the mass(es) of the dunes which result from
the interaction. To do this, we study the numerical results of many, many interactions
(as discussed in subsection 5.2.1), which results in construction of a 1-to-1 interaction
function (f) between the before-collision size ratio (r = (adownwind/Aupwind)before) and
the after-collision size ratio (s = (adownwind/Aupwind)after) (Figure 5.5). Note that
r ∈ (0, 1), but that f(r) = s ∈ [0, 1), as if coalescence occurs s = 0.
We find the following expressions (where lowercase always denotes the smaller of
the two dunes):
(aafter/abefore) =1 + r
1 + s
s
r(C.8)
(Aafter/Abefore) =1 + r
1 + s(C.9)
Thus, if coalesce does not occur (i.e., s �= 0), for a dune of mass M∗ to result,
we need an input dunes of size abefore = M∗
rs
1+s1+r
, Abefore = M∗
1+ss(1+r)
or abefore =
M∗
r(1+s)1+r
, Abefore = M∗
1+s1+r
. If coalescence does occur (s = 0), then to get a dune of
mass M∗ to result, we need abefore = M∗
r1+r
, Abefore = M∗
11+r
, and r less than the
153
coalescence threshold (= 1/3 in our example, Figure 5.5). These expressions allow
us to fully define our δ-functions, as they uniquely define the relation between the
pre-collision and post-collision dune sizes.
154
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